Keep ALL oil or cleaners away from the firing pin hole and surrounding area!!!
It can easily leach into your primer pocket and deaden it, resulting in a "click!" at the very WORST possible time.
The way I do it is clean briskly with CLP and a toothbrush in that area, then do the rest of the gun. When I'm finished, I come back and scrub the area around the pin as dry as possible with clean patches, and carry it a step further by using a pipe cleaner to dig into the pin hole and clean out as much residue as possible.
As I type this, I realize I haven't answered your question!
The answer is YES, clean the area, VERY well. Then, DRY, the area, SUPER well.
This was a good question; glad you asked it.
Flyer