Possible HIV exposure

So I had a medication non compliant HIV patient who I was doing blood work for last night. I don't know how, but it completely escaped my mind to be more vigilant around this patient's bodily fluids. I degloved one finger to palpate the vein and when I retracted the needle, I put some gauze over the puncture site and saw some blood seep through the gauze before putting tape over it. This blood may have accidentally gotten on my bare finger. I then hand sanitized my hands, except I noticed a cut on the base of my thumb (which was covered by the glove).

I'm not sure if this counts as an exposure. I just worry that the hand sanitizing spread the virus around my hands and potentially allowed it to make contact with the cut. I'm not even sure if hand sanitizer kills HIV.

Looking for opinions. Thanks.