given that you have only one wire from the new light, it’s probably fair to presume that your new light is self grounding by a screw through one of the two mounting holes.
if I was doing this, I’d first pull the black grommet out of the skin and see if I found a splice in the existing wire, which I’d then cut away.
then I’d clip the plug off the new light, slide that grommet onto the black wire, and then connect the new light with one of these connectors - SolderGrip Heat-Shrink/Solder Multiple Wire Butt Splices, 22-18, 15/Clam-CPGI-CWT-9002-15 at The Home Depot
(they also come in larger sizes for 16-14 and 12-10 wire)
then I’d stuff the splice back into the skin, closing the hole with the grommet, and screw the light down.
am curious to ask what manufacturer & part number of the lights you’re using. please also let us know how they look when done?