As I was checking my finished rounds, I noticed that some of them had the primer protruding slightly. Can I set it in further by using my priming tool?

Yes. I've had to do that almost every reloading session once or twice. Put a little more pressure on it and test with your thumb.

Try reaming out the primer pockets before you apply more pressure.

It mostly happens with S & B brass. Their primer pockets are tight and just deep enough so that the primer fits flush.

Do you clean your primer pockets?
My normal case prep routine is I first size and deprime cases, put in tumbler for an hour with walnut shell media, remove and examine each case for media in the flash hole and clean the primer pocket then finally trim the cases if needed.
I made a simple tool for cleaning out the flash hole and the pocket. I took an old screwdriver and with a file I removed material that resembles a small paddle bit. The tip fits the flash hole and punches out any media and a simple twist removes carbon fouling in the primer pocket.

Finished rounds I think I’d pass on trying to set those deeper…

This isn’t an instance where giving it another “uggadugga” or getting a bigger hammer is the solution. But if you check your priming system seating depth, then reseat with normal pressure, being very careful to go slowly and avoid slips which would crash the tool - all of the same caution you use when typically operating the same device with the same potential for explosive results, but acknowledging the magnitude of consequence is increased, then reseating primers on loaded ammo is a reasonable task.

If the primers resist seating, then it is indicative of some pocket problem. This should be caught in the primary loading process, so this increased consequence profile isn’t part of our life.

if it were me id use a Lee hand priming tool. this way you wont assert too much pressure on the primer.

Nomercy is again correct. But the OP never said if he was using a single stage press or progressive. Either way the primer seating depth should have been checked right off.
I don't use progressive presses so I'm not familiar with how or if you can adjust primer seating depth. I seat all primers with a RCBS hand priming tool and check each one for correct depth. And my primer pockets are all cleaned prior to priming. Some pockets are really tight and take quite a bit of pressure to seat the primers. Sometimes rotating the case in the tool helps.

Bronko, im the same as you when priming my casings. i use a Lee priming tool and always run my finger over the primer after repriming a case. this way i can tell if the primer is seated in far enough. i make sure all my primer pockets are cleaned before priming. ive found that using the hand tool for priming, the case wont come out of the shell holder if the primer isnt seated deep enough.