Some ideas.
Chances are that the coring has rotted and there is nothing for the screws to grab. You could try some epoxy in a hypodermic needle but I think you will find that the hole never fills. If that is the case you might try drilling 1/2" holes and then epoxying a 1/2" dowel in the hole. Redrill for the screw.I think you should scrape the bulk of the old caulk off and then not worry about it. Because the portlights are a plastic you have to be careful with any chemicals like mineral spirits.The caulk has to fill the gap between the spigot(frame) and the hole. It should be heavily applied and squeeze out the outside. In other words no water should get under the trim ring from the outside. I epoxied the perimeter of the holed before I installed the port. I know friends who did not have anything to epoxy, the coring had rotted completely away.There is a chance you will find the same with the handrails, that pouring epoxy into the holes never ends. Best bet is to drill the holes out to about 3/8" and fill with an epoxy mix like peanut butter. Use lots of blue tape or you will have it all over the non-skid. Unfortunately the handrails are just one source of water in the cabin trunk on a sailboat. There are the hatches, the companionway and cover, dorade boxes, and the mast partners or step. I also have winches, clutches, cleats, deck organizers, and a traveler(H37C). Great fun.