2nd January sees the Earth at perihelion – the closest point to the Sun in its orbit. While it might make a little difference to the weather, that difference is not so much as the effect that day and night and that the seasons have.
The night 3rd/4th sees the maximum of the Quadrantid meteor showers (shooting stars) and should be reasonably visible round about midnight after which the waning Moon appears.
Mercury might be visible just before sunset at the beginning of the month but it reappears in the dawn sky at the end of the month.
I sometimes complain about the light pollution which spoils our view of the night sky in our neck of the woods. However, it is worth remembering that it takes some time for our eyes to adapt to low light levels – typically about 20 minutes to get our ‘night vision’. So we need not give up too quickly. Wrap up warm and try to find a sheltered spot out of any wind and not lit by street lamps or security lights and a be prepared to be patient. Of course, if it is cloudy, no amount of human night vision is going to make much difference!