To answer my own question here, it seems that I am correct.
mfranjic has previously posted this database of technical details.
It shows that most Class 800x trains have one traction motor per axle rated at 226kW and all four axles are powered in each motor vehicle, meaning each motor vehicle has a total power output from the traction motors of 904kW, or 1,212hp. This means that on a typical 5 car unit, the total power is 2,712kW, or 3,637hp. When on diesel, they have either 575kW per engine (787hp) in the case of LNER, or 700kW (939hp) per engine in the case of the uprated units operated by GWR, Hull Trains and Transpennine Express. Therefore, the total power output from the engine is a maximum of 2,100kW, or 2,817hp. Of course you then have to deduct the auxiliary power for the air conditioning, lighting etc, plus there will be losses from the engine as heat etc. Therefore the power available on diesel is far less than when running in electric mode.
The Class 810's are different, however, as only two of the five coaches have traction motors. However, the traction motors are more powerful, at 290kW each, or 389hp. Therefore, each set has a total power output of 2,320kW, or 3,111hp. The unit has four diesel engines that are also slightly more powerful, at 736kW (987hp), and therefore the total power output from the engines is 2,944kW (3,948hp). Unlike with any other Class 80x unit, the installed diesel power exceeds the installed traction motor power. Some of this power will be used to power the air conditioning, lighting etc, and some will be lost as heat, but it seems these units are designed in a way that the diesel engine can provide enough power to allow the full output of the traction motors, or at least very close to it. They are less powerful when running on electric mode than the other Class 80x units, but do the existing trains use full power anyway? I've seen mention of DfT specified acceleration curves being lower than full power. It's also still a healthy amount of power for a five coach train. By comparison, a Class 222 has 559 kW (750 hp) engines, so a five car unit has a total engine power of 2,795kW (3,750hp).