If the socket itself is broken (cracked) there is a higher likelihood of parts shifting around and wires shorting out. I would want the socket removed or replaced. Tape is not a reliable solution in this case. Wasps or other insects could be attracted to a remaining void.

Either tape is fine but neither weathers very well outside. Duct tape would last if you're going to fix within the year. Those fixtures are inexpensive. You can even get an LED version that lasts much longer and doesn't use as much electricity.

Not sure what a DH is.

I would replace the fixtures if theyre that old that theyre falling apart.

I guess I could just make sure the switch for these 2 lights is always off. Would that keep it all safe? I can't turn off the breaker to it, since that breaker keeps lots of other things on too.

Dave....DH is short for "dear husband". :)

Sorry for the delay everyone. thanks for your responses. For some reason, they just showed up today. Thanks again.