How would I replace this recessed light with clips?
It could be that you pull straight down.
Or it is more likely to be a gu24 base that you turn and pull down.
A GU24 base has two pins that you insert into slots and then turn to secure. Usually you have push the bulb into the socket as you turn it, both installing it and taking it out.
It appears that the bulb is broken at the top. Where gloves, eye protection and a mask when removing it.
image of an adapter with a GU24 base from amazon.com
Image from Grandbrass.com