Can you make the cut of a player conditional on the signing of another? Thanks

That's on the feature list, but unfortunately it's not there yet. The best way to handle that in the mean time is to sign the player on the midweek and then cut the other player before the game sim. It's also unlikely that the player you are targeting will reject your offer, unless he just got cut from another team in the most recent stage.

Also note that if you sign a player during a game stage, he will be added to your roster as an inactive, but the AI will reset your depth chart if you have fewer than 45 active players.