Copied from the MN regs, “ A person on foot may, without permission, enter land that is not posted to retrieve a wounded wild animal that was lawfully shot, but may not remain on the land after being told to leave.” That’s the short version you need to read the regulations as they have a section on trespassing in the booklet. Or call your local officer to answer questions.
I think this law is mostly intended for deer hunting. Although it does not specifically indicate what the "wounded game" could be, my general rule is that if its on private land and I do not have permission there, I gain permission before entering or do not go there, period. If I had shot a turkey, pheasant, or deer and it was wounded and entered private land that I did not have permission to be on, I'd do everything in my power to gain permission and retrieve that wounded bird/animal. Most land owners, even if they do not want you on their land, understand that leaving a wounded animal/bird is not very moral and they will allow you to go in there and at least make an attempt.
If you see the dead animal laying out on the ground or the wounded animal is not able to flee than I would leave my gun at the property line and go pick it up or have the dog retrieve it. Law is clear and you are only on that person's land for a minute or two. You should be mentally prepared that you may be watched and that the land owner may call the sheriff, GO, or chose to confront you later when you return to your vehicle.
If the animal is wounded and flees relatively able to keep moving or a bird lands in a cattail slough on the other side, then following gimruis' advice is the right thing to do.