No, it's an illusion with no speed. Difficult Terrain does not affect it.
The Trickery Domain Cleric's Channel Divinity: Invoke Duplicity states:
As a bonus action on your turn, you can move the illusion up to 30 feet to a space you can see, but it must remain within 120 feet of you.
You are moving the illusion, it does not have a speed. Having a speed is required for Difficult Terrain:
You move at half speed in difficult terrain--moving 1 foot in difficult terrain costs 2 feet of speed--so you can cover only half the normal distance in a minute, an hour, or a day.
But does anyone notice?
That part is going to be up to you as the DM determine. There is no rule about this, but do note that there are many spells/abilities/magical items that let you avoid difficult terrain speed changes. Whether or not that's an impact on the observers is up to you, but you can either choose to have them ignore it, use passive perception/investigation, an active roll (which may or may not take up an action), or something else. Heck, it could even be just a simple Dash to cover the extra terrain (thanks Ruse!)
But be consistent and fair. This is a limited use option with their Channel Divinity and you should take that into account before nullifying it. If you do plan on doing this, I would recommend letting the Cleric know before hand that's how you plan on adjudicating and give them an opportunity to move their illusion the distance they'd like.
Speeds are not all equal
It's important to note that the illusion can only be moved up to 30'. This is regardless of the actual speed of the Cleric. They could normally have less movement (like a small creature), or more movement. They could be under haste and have a doubled movement. The 'obviously ignoring difficult terrain' is neither obvious nor clear that it's because it's an illusion.
Lack of rules around discovering the illusion
Unlike spells such as minor illusion, major image, etc. that provide mechanics for uncovering that it's an illusion - Invoke Duplicity does not.
minor illusion: If a creature uses its action to examine the sound or image, the creature can determine that it is an illusion with a successful Intelligence (Investigation) check against your spell save DC. If a creature discerns the illusion for what it is, the illusion becomes faint to the creature.
major image:Physical interaction with the image reveals it to be an illusion, because things can pass through it. A creature that uses its action to examine the image can determine that it is an illusion with a successful Intelligence (Investigation) check against your spell save DC. If a creature discerns the illusion for what it is, the creature can see through the image, and its other sensory qualities become faint to the creature.
Mechanically, it's always on and always works. Of course, someone attacking it would realize that nothing happens, and they may not attack it again. But the mechanics around being within 5' of it are still active:
Additionally, when both you and your illusion are within 5 feet of a creature that can see the illusion, you have advantage on attack rolls against that creature, given how distracting the illusion is to the target.