It depends on the well pump, which is why this question cannot be answered with a yes from a website. Honda’s home table does not list well pumps. A 5,000-watt (running) generator has a hard running ceiling of 5,000 W, and a starting (surge) rating that is often only somewhat higher, depending on the machine. An Energy Star refrigerator in Honda’s table is 192 W running and 1,200 W starting. A labeled 400 W of lights is 400 W. That leaves roughly 4,400 W of running room for the pump in this arithmetic — which is not the same as saying a typical well pump fits, because starting current may not.
Use the pump nameplate: running watts, or amps × volts, and starting watts or locked-rotor amps × volts. Add the fridge and lights, then add only the largest starting increment to the running total, the same method as the well-pump-and-refrigerator page. If the pump is 240 V, a 120 V-only generator will not run it correctly.
If the nameplate is missing, stop and find it on the motor or in the manual rather than borrowing a forum “half-horsepower” chart. Enter the loads in the calculator. A 5,000 W generator is also not automatically enough for an electric dryer (Honda: 5,400 W running) or for central air.
This is not a permit calculation, a utility approval, or a substitute for a licensed electrician, HVAC contractor, or the authority having jurisdiction. Panel Headroom is an independent educational site. It is not a utility, a certification body, or a contractor. Electrical and generator work that is done incorrectly can cause fire, shock, or carbon monoxide poisoning. Use the adopted electrical code and manufacturer instructions; pull permits where required.