In le mans, they run their fastest laps during the night. Michelin brings 3 compounds of slicks and to counter the fact that the tyre becomes harder to warm up, most teams switch to a softer compound as the sun sets and track temp drops. In F1, they can't really do that, and it can be a problem. Track temp dropped a bit between Q2 and Q3 in China this year, suddenly everybody was running 5 tenths slower that they were in Q2.
If temp drops too far though, there isn't much you can do. I remember being at a formula student competition where the track temperature was around 7 degrees C. I can't remember whether anybody was running the Hoosier LC0 compound (which to my knowledge is the softest available FSAE tyre) but people were struggling with poor traction all day. the front would wash out on turn-in, then the back would kick out when they tried to get back on the power. Lap times plummeted when the track warmed up a bit later in the day to about 13 degrees C. This is what happens when you try to run a formula student event in Canada.
So as for how temperature drops affect tyre performance, it cuts both ways, but it is definitely significant, probably much more so than the effect on engines in most cases. Besides, in most cases, the engine plays a relatively small role in lap times provided its running reliably. the corners is where you gain time in almost all racing series; the only obvious exception is drag racing, and even then, getting the tyres to work perfectly so you can get a good launch is crucial. Also, if you had a more powerful engine, you wouldn't put that much more energy into the tyres. You would heat them up a bit better under braking, but the changes in power that we are discussing right now would have a very small effect.
Finally, remember that there is a big difference between ambient and track temp. On sunny days, the delta can be on the order of 15 degrees C.