My first time shooting a small frame pistol.
As a general rule, smaller/compact/subcompact pistols will produce more snappy/jerky recoil compared to larger/full size pistols.
This Hellcat is stinging my trigger finger ... Is it me, or the gun ?
What you are experiencing is inherent to shooting small/compact/subcompact pistols. Many people who shoot "pocket" 380Auto complain of similar.
The trigger "slap or sting" is from recoil causing sharper muzzle climb that is also rotating the trigger upwards and putting pressure on your trigger finger pad.
Difficult to hold firmly because it is so small. Maybe that is my problem, not sure.
I shoot Glock 27 40S&W and M&P Shield 9mm, both small subcompacts and with factory loads, recoil is definitely more snappier than full size G22 or M&P40 but I do not experience trigger slap or sting.
But with Taurus TCP 738 chambered for 380Auto, different story. Even with locked breech barrel and dual recoil springs, there's only so much you can do with such a small package to attempt a gradual slow down of slide and slide to frame slap will induce muzzle snap and I can definitely feel it in my hands and trigger finger. To tame the snappy recoil, I worked with my reloads to produce milder recoiling practice rounds that were still accurate -
https://www.thehighroad.org/index.php?threads/380auto-x-treme-100-gr-rnfp-range-test.748320/
At 7 yards with TCP 738, factory felt recoil designated 10/10, 2.6 gr load produced 4/10 felt recoil, 2.8 gr 6/10 and 3.0 gr 8/10 that were much more comfortable to shoot for extended range sessions.
So if you reload, one option to tame the snappy recoil and trigger slap/sting is to load lighter target loads for range practice.
Another suggestion is reviewing the shooting basics of improving your grip -
https://www.thehighroad.org/index.php?threads/flinching-drills.864546/page-2#post-11416785
After working on improving your grip mentioned in above linked post, consider working on "mental" aspect of shooting illustrated in this post which may help -
https://www.thehighroad.org/index.php?threads/flinching-drills.864546/page-2#post-11419944
You can also try shooting other small/subcompact pistols to see if you experience the same trigger slap/sting and if you find that only your pistol causes the trigger slap/sting, then you can consider replacing the pistol.