[Schroedingersat]

The point isyou get x units of heat and y units of electricity (probably around 2 and 1) from x + y + a tiny bit units of chemical potential.

You can't put x units of heat that is already hotter than the hot well into the cold well of an idealized heat pump and come out ahead of just putting it into the hot well.

You could run it through a heat engine (TEG or stirling engine or similar) and extract some work with your heat pump's 'hot well' as your 'cold well' then use the work to move other heat inside but this is extra complexity.

There might also be eg. some temperature of your cold well where your heat pump's efficiency diverges because viscosity changes or you're not in a situation where modelling it as a single phase change in your working fluid is good or something. Then it might help to waste some free enthalpy (but not discard the energy into the environment entirely) to warm it up slightly.

Presumably none of these things are worth the added hassle especially if x + y is already close to the heating energy needed and y is close to the electrical energy needed inside. The benefit of whatever convoluted scheme you come up with would only be COP * second_heat_engine_efficiency * x which is likely around the same size as x or a bit smaller. Maybe a 30-50% reduction in overall energy input for more than doubling the complexity.