It really depends on the type of cut.
If you’re taking over a new team or making a cut before the keeper deadline, those cuts are penalty-free. In those cases, the player’s new minimum salary is $1.
If you are cutting a player after the keeper deadline and you’ve already used your penalty-free cuts, there will be a cap penalty for cutting that player. That cap penalty is half the player’s salary, rounded up. That cap penalty will count against your cap and will be the player’s minimum salary at auction. Once the player is signed, the cap penalty goes away.
During the season, you cannot re-acquire a player you cut in the last 30 days. This is not true for the auction draft. So if you do end up cutting a player with a cap penalty before the auction draft, you can remedy that by starting their auction and bidding.
Generally, teams do not go into the auction draft with cap penalties, so this rarely comes up. It did come up tangentially in this thread about the keeper deadline, which is worth a quick read.
Hope that clarifies!