What phenomena can explain not detecting a planet while detecting a more distant planet?

The habitable "planet" is actually a moon of a super Jupiter. Telescopes in the Sol system were unable to resolve the moon from the planet it orbited. However, as the colony ship passed much closer to it on its way to the planet that was detected, they got a better look and spotted the moon.

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An lengthier answer in response to comments:

The question doesn't give any information about how fast the ship travels, just that journey time is 80 years. Lets assume a distance of about 20 light years. That makes the ship fast, but not completely unrealistic with near future tech. If they check every star system within that distance, that's a survey of 83 star systems. That feels like a lot of work to do in just four years.

The search was presumably for habitable bodies. Exactly how they orbited the star would be of less importance than having surface water, suitable atmospheric pressure/composition, comfortable gravity, protection from solar radiation and so on.

However, while detecting exoplanets is really hard (See Ash's excellent answer for a summary of the techniques used), detecting exomoons is even harder. As of October 2021 I don't believe that a single confirmed detection has been published, although there are several good candidates.

Given the difficulties and time pressure, it doesn't seem unreasonable that researchers might look at a system and say "There is a large Gas Giant that makes a Terrestrial planet in the habitable zone impossible... Lets move to the next system." Missing the tiny hidden signal that gives away the presence of the moon...

The searchers pick their (more distant) target and launch the ship. By good luck however, the star system with the better habitable body is close to the trajectory to the target star. Passing within a few thousand AU, the difficult to spot (at 10ly) moon sticks out like a sore thumb to the ships sensors.

Have you ever walked in a fog so thick you don't see a pole until you slam your nose onto it?

This is what might have happened: if in the line of sight between Earth and the planet there was something increasing the noise, the signal of the planet presence would simply become too small to be detected.

It could be dust, it could be gas, it could be a more luminous star.

It wasn't visible from the original angle:

Since you're talking about distances in 10s of years, I have to assume the planets we're going to are extrasolar (that is it orbits a different sun than our own/their own).

If it has a very long orbit (Pluto take 247.78 years to orbit the sun), then perhaps it was just behind the sun the whole time! Once you got far enough around the sun, the slowly orbiting planet was revealed.

If they're inside the solar system, the timescales are a bit off I think, but you could do the same concept, where the orbits are synchronyzed with the planet always hidden behind our sun.

The simplest answer: the star, and it’s planets were not visible from earth because a nebula was in the way.

Nebulas are clouds of gas and dust (mostly hydrogen) that obscure visible light. We don’t really know what may be on the other side of one and indeed something like half of the Milky Way is not visible to us because of obscuring nebula. So we can actually see more of the Andromeda galaxy than we can of our own.

Here’s a famous example, the Pillars of Creation

In this image, the stars that look like they might be in the nebula are actually in front of it, we do not know what is behind it. However, a spaceship that had moved far enough away from the earth, could see what was behind it, including perhaps, previously unknown stars and their planets.

Here is an example of a true dark nebula, Barnard 68

Like all known dark nebula it is large and far away. However, this does not mean that all dark nebula are large and far away. In fact there is every reason to believe that smaller obscuring clouds of gas and dust (i.e., dark nebula) are very common, however, the problem is that like Barnard 68 above and the vast majority of dark nebula we cannot see them, we can only infer them because of what we cannot see. So we can "see" Barnard 68 only because of its size and the large bright star field behind it. (the Pillars of Creation nebula is one of the rare exceptions, we can truly see it because it is lit up by the stars just in front of it).

For smaller nebula that are not in our line of sight within the galactic plane, there are neither enough stars behind them, nor are they large enough to allow us to "see" them in this way. However, because stars are created from nebula, there's every reason to believe that nebula are quite common.

I make no claim to sufficient expertise in astrophysics to assure you the following is credible... but I did sleep virtually (given a bit of literary license) in a Holiday Express last night, so you're in luck

OK, I admit it, I slept very comfortably in my own bed last night. Still...

• Parallax: the effect whereby the position or direction of an object appears to differ when viewed from different positions, e.g. through the viewfinder and the lens of a camera.

• Gravitational Lensing: A gravitational lens can occur when a huge amount of matter, like a cluster of galaxies, creates a gravitational field that distorts and magnifies the light from distant galaxies that are behind it but in the same line of sight. The effect is like looking through a giant magnifying glass.

• The second planet orbits a white dwarf, a star that is hot, but dim.

• The second planet orbits a star that dims oddly, meaning that there's some unknown effect causing the star to be dim at the time the first planet was discovered, so the second planet couldn't be easily detected. Examples of what theoretically could cause the dimming would be a massive dust cloud in orbit around the star or a massive planet in a closer orbit that orbits at one heckava inconvenient orbit.

• To build on @L.Dutch's answer, a dust cloud or nebula that masks the view of the second planet from Earth, but the point of view changes as the ship travels, allowing it to see behind the dust cloud or nebula.

• The planet orbits (very close orbit, here!) an unusually warm Y Dwarf star that's too cool and too dim to allow the planet to be seen from Earth, especially with that whomping close orbit....

(this answer assumes sub-lightspeed travel, say less than 30%)

They came too close, the cloaking does not work anymore

The newly discovered planet has an advanced civilization. Unfortunate events in the past were a good reason to develop a planetary cloak. This cloaking technology prevents the planet to be discovered by pirates roaming around the section at warp speeds. It is based on a trick involving satellites and light interference patterns.

This cloaking field has a certain distance threshold. When you come near, at some point you'll not see the interference patterns anymore, you see the planet. The colonists have come really close. Of course their approach was discovered by the aliens years go.. it was no reason to change the cloak, these primitive humans pose no threat..

Trojan planet

Trojan planets, which follow or precede the lighter of two stars at a 60-degree angle (L4/L5 points), can usually be detected by transit timing variations. This is important since the Earthlings would not have launched an interstellar-level project without some very good research, possibly including unmanned observatories flying several light years into space and measuring transits not visible from Earth. However, the paper cited also shows (Table 5 at the end) that the transit timing variations can be extremely small for a few cases. So there is a point at which astronomers could have written off the odds of a planet in a given system based on a limited period of instrumental data availability. Coming nearer the system, observing it with the best available instruments, the ship might detect more subtle variations, or brute force an image resolving the planet by eye, or use some cleverer astronomical deduction.

Consider the real-life case of trying to see planets orbiting our nearest neighbor: the binary stars are too close together so the glare spoils the effort.

The star in question could be a binary, or an "optical binary" where another star is in our line of sight behind it. Either way, having another star in the field where advanced telescopes carefully block out the image of the central star to try and see planets — and also obtain the atmosphere spectra of the planet — would prevent such an observation.

Perhaps we knew there were planets there. What we didn't know was the exact mass and radius and "life signs" primarily gained through visible direct observation of that planet.

Years later, such measurements became possible due to further advances in technology, or the movement of the stars out of the way, or observatories that are stationed far away from Earth.