Generators are poor enough on efficiency, because you may need a 5 kw generator to meet all your loads, but the demand hovers between 100 watts and 5kw. At lower current draws, the generator becomes ludicrously inefficient.
A modification to this is to use a bank of batteries and inverter. Use a smaller generator which can run flat-out all the time charging the batteries.
But if you have a heat pump, depending on the system, the start-up load can be very high, out of the range of most small generators and inverters. This info should be on a plate somewhere on the system.
If you use a generator long term, using a water cooled one, and retrieving the waste heat might be an option.
Lastly, there are small gas fired CHP units coming on the market which can produce up to about 1kw of power. Once they are running, they can be kept going supplying power up to a maximum of 1kw for lights, telly and other "essentials". These are very efficient as all the waste heat is used, but perhaps not in summer time...