You can customize your engine on chess.com to change how far it calculates. It may require having a premium account
chess app or engine with customizable analysis depth

Can anyone point me to a chess app or engine that can limit the computer to analyze only a specific number of future moves? I'd like to see if it can help improve my own ability to "see" the pieces on the board. Thank you!
If you're auto analyzing a game, either Lucas Chess or Arena will let you specify the depth of the analysis.
If you're trying to manually analyze a position, I think it's harder to find something to do that. The only way I know to do that using Arena is to install the old Crafty engine. It's a Winboard engine, and it has its own configuration file (crafty.rc). You'd have to add the "sd" command to specify the max search depth in plies. For example: sd=6
For manually analyzing a position In Lucas Chess, I think you can configure the tutor engine for a particular search depth. Then when you're at the position of interest, click on the "Help with Move" button.
https://lucaschess.pythonanywhere.com/
Here you can get the crafty engine (crafty25.3.exe) and config file (crafty.rc contained in the craftyrc.zip file): https://github.com/MichaelB7/Crafty/releases
The help file for the crafty commands is harder to find - I'll have to look for it. :-)
Ah, the crafty commands can be found in the file named "crafty.hlp" located in the source file download.

Fritz by Chessbase allows many analysis settings including selecting how many first moves, second moves and third moves to analyse and to what depth of ply or maximum time to spend on each move.
No need to be forking out too much hard earned on the latest version either, look out for a used copy of Fritz 12 or later on the internet auction sites. You will not be disappointed
Can anyone point me to a chess app or engine that can limit the computer to analyze only a specific number of future moves? I'd like to see if it can help improve my own ability to "see" the pieces on the board. Thank you!