Yes you may get a little more power on a cold damp day,
You get as much power as you need to push the car along. The power vs air density curve may affect racing performance, where max power makes a difference. But not for day to day driving.
There are two things going on: Friction (engine and tire) vs temperature and engine efficiency vs temperature.
Air drag (like lift) goes down as temperature goes up. Engine friction is probably relatively constant once warmed up (being dependent on the thermostat set point). But engine pumping losses are lower for less dense air (warmer). And this probably affects diesels more so than petrol engines given their 'wide open throttle' operation.
The other factor is the additional energy needed to heat colder air up than warmer air. Thermodynamics isn't my specialty, but given displacement and RPM, the mass flow and resulting cylinder PV curves would reveal the work difference.
