Customizing the smallest cog I am running on a Shimano Tiagra 10 speed (CS-HG500-10) (12T-28T) and (50-34T front). I'd like to change the smallest cog (12T) to something smaller, i.e., 11 or even 10, to improve my sprints. Is that a possibility?
Yes you can.
You could need a new cassette screw too, the current one you have will be too big for the 11T cog.
Realistically, the 1T different will not make as much a difference as improving your techique, muscle/power, tyre and air pressure choice and even nutrition.
Make sure the new 11T cog is for Shimano 10s cassette, it might not fit otherwise. Worst case scenario, you can get a complete new cassette, Like a 11T-25T or 11T-28T etc. Plenty of brands available with them.
I don't think you can get a 10T cog on it.