Yes, there are a couple of things I can think of. But the help you better, answer the following:
1/ Why did you need to replace the screen? Is the phone a-water damaged, b-screen smashed or c-another incident? Which is it?
2/ Do you know for sure that the home button did not have this issue before the screen replacement?
3/ Was the phone repaired before? If yes, why?
Now here's a trouble shoot you can start with:
Option 1: if your home button is clicking when the screen is lifted from the frame, BUT not clicking when you fit it and screw it in place, it could be that something is pushing on the home button from the inside and/or the frame of the phone is too thick (manufacturing defect), causing the home button to be alway pushed only when the screen is fitted in place. Try a new screen or look for something that might be pushing up.
Option 2: pry damage. Remove the screen, disconnect the battery, and use magnification in the order of 10x, 20x or more if possible to examine 2 specific areas:
1/ Between the battery connector on the board and the shield located to its right. You can see a tiny strip of the logic board, and it has a few tiny components. Is any one of them missing, damaged, or part torn from the board? If you suspect one of them is part torn off, try to poke at it (very very gently) using a needle. Don't loose it if it's coming offf. Then more generally look at all the components around the battery connector area. Anything suspicious?
2/ Examine also the logic board around the screen and camera connectors, especially the top part of the area. Remove (VERY VERY carefully) the black "foil" that masks the components, and look for anything missing/damaged/part torn off the board.
Let us know if you find something missing/damaged/suspicious.