How can I fix my cold water tap which works on and off?

Oh no....Are your pipes freezing? You need to put immediate attention on this when it happens. You want to get out a hair dryer, and apply heat to the faucet area....and if you can get behind that wall where the faucet is to get closer to dethaw your pipes before they burst! At some point you need to find out why they are freezing. We were having this problem when we first moved in our house. Our bathroom was above a porch(like an overhang) and the wind would whip up right on that side of the house. My hubby had to apply more insulation behind the wall and we eventually had to put a new tub in to apply more insulation underneath the tub to protect it from the cool wind. This fixed the problem. But for a temporary fix, keep your faucet running just slightly like a drip. Running water won't freeze as easily. Good luck!

https://www.youtube.com/watch?v=tpsRjvnuO0w

I agree with K. It sounds like your pipes might be freezing, which is a really serious problem. She gave you some good advice. I also had a problem with pipes freezing, and it was because we were using our pellet stove too much, rather than the regular baseboard heating. The pellet stove heated a localized area in the house, rather than evenly heating the whole house, and the pipes running through the walls farther away from the pellet stove froze because they weren't getting enough heat. If you have a wood stove/pellet stove/fireplace that you use to heat, make sure you're running your furnace as well to heat your home more evenly. If you have your heat turned down below 65 degrees, turn it up to at least 65. And, like K said, you may have to add more insulation, but these are relatively quick and easy fixes you can try in the meantime.

Could be bad washers in the cartridge. Loose or not seating properly. Different brands have different ways to replace. Single or double faucet. Could also be debri in the supply pipe that floats and sometimes blocks the passage.

https://www.google.com/search?ei=1V7-Xf-gDdLbtAbgz6kQ&q=replace+faucet+cartridge&oq=replace+faucet+catr&gs_l=psy-ab.1.0.0i13l7j0i13i30l3.135092.143266..148943...1.2..3.131.4617.30j17....2..0....1..gws-wiz.....6..0i71j33i22i29i30j0i362i308i154i357j0i273j0i131j0j0i67j0i22i30j33i160.9t1Ej1yjsoE

Does you faucet have 2 handles or 1 handle. The first thing I would check in the easiest to check would be the aerator the little round thing where the water comes out . they usually have screens in them that can get clogged up. hard to say without looking at your faucet but if you have a two handle faucet is probably either the supply line going into it or the faucet stem " diverter" inside the faucet handle itself. I would shut the water off and take of the supply hose that conects to the bottum of the faucet , hold it in a bucket then turn the water on.they often will get clogged with sediment as well. The stems in the handles are usually under $10 if it isnt the supply lines or the aerator. if it's a single handle faucet the cartridges can be a little I see if it's an older one and they have a ball and Rings type set up it's just so hard to say without seeing your faucet but it's all fairly affordable cartridges can be a little pricey but we're talking 30 bucks or something hope this helps

Sounds like your stem is going the stem is what holds the knob on it needs to be replaces.

Could be your tankless or water heater theres a few things like annual descale or maintenance should of been done on those units .the same with water heaters they require flushing. Call a professional plumber and get 2or 3 estimates