Ryoko how are you today? If your DF/Dp won't let you put heater on, he is wrong.
Some people on here are being harsh Ryoko. It is because they are concerned about your dc. They don't seem to understand harsh words will drive you away from looking at posts that have kind words and useful advice.
If he pays internet bill, and is controlling, stopping internet or other money saving ideas will not give Ryoko access to money for heating.
I'd say give dc a quick bath, cold won't hurt him for a few minutes. If you really feel you can't, at least clean skin creases with baby cream. But please just give dc a bath.
If you decide to leave, or kick him out, you can get help with money while you look for a job with better contract and hours. And you can get the CSA to claim money from dc father to support DC.
Can you phone or go to a Sure Start Centre, as someone else suggested? They can help you and also refer you to other sources of help.
Please ask for some real life help. If you really feel you cannot go out, phone for advice, look on internet, or ask for a home visit. Don't say you need home visit because waiting for a parcel though! That will lose sympathy!