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I have an L-shaped cellar, and the rooms in the leg are in reasonable nick ventilation wise - There are a few airbricks which seem to keep an airflow. The room at the foot of the L, though, has no airbricks and gets a bit moldy. There is a broken extractor fan which has been severed from power.
If I replace this, does it really need to be connected to the mains / light switch like you see in bathrooms, so it comes on automatically when you switch the light one? Was wondering if you could just get one with a plug and switch it on as and when, or if that is a rubbish idea. Alternatively maybe I could just bin the fan and put some air bricks in?
Picture of sad-looking extractor - room is medium size, ca. 5 x 5 m.

PC cooling fan and a PV panel.
Maplins would have been the place. Internet?
APF
THks Alex - hadn't considered a PV panel, unfortunately the wall faces out onto my neighbour's property so might not work for me. Think I need to power it from inside my house.
You can put the panel where you wish, just a longer cable.
A
I used to have a big cellar that got a bit damp. It wasn't sealed and the ceiling height was too low to bother spending a wedge on it - tanking it would have only made it shorter!
Solution. I did as you thought and replaced and installed a small but good quality fan that was wired straight to the mains and was left on pretty much 24/7. I also dropped down pipes from the radiator above (in the front room) and put a small rad in there as well which helped a lot. I did then use a dehumidifier as well, but didn't need to after a week or two once it all dried out. Job done.
Caveat. Cellars very dramatically, however I'd certainly start with replacing that fan and getting it going. That will help move the very probably damp air out.
Ive got the same issue in a house my parents are the landlords of. We’re currently renovating so have taken the chance to tackle it. Solution has been to seal it off well from the rooms above, blocking any draughty bits and such. We’ve then installed an inline fan wired straight in to the mains, controlled via a switch in the room above. The fan is screwed in to the joist and vented via a flexible hose connected on to a section of soil pipe. The soil pipe goes up out in to the garden and has a pest guard over the end of it.
The advantages of going with the inline fan are that you’re not as restricted by the size of the hole in the wall (150mm dia fan in ours) it’s easier to access, wire up and replace if it dies and the bigger fans seem to be more reliable.
When the house is ready to be let out or rented again we’ll probably shift the fan from the joists as there is a bit of noise transfer, nothing crazy though.
Make sure your air bricks are clear, not just visually from the outside but all the way through. They’re supposed to have a duct through the wall if it’s a cavity one to make sure crap doesn’t block them at the back.
Replace with similar fan with humidistat? Not sure the level of humidity they need to trigger though.
Not a cellar, but back of house is partially underground with a back room coal hole / store. This room gets a bit damp, so did the same as Hugo, routed in central heating with a small rad without TRV (ie is always on whenever CH is on). Put a dehumidifier in and set to economy auto, dried it out quickly and hardly activates and/or needs emptying anymore.
Don't see why you couldn't just replace it with an RVK or similar fan that's mains powered if it's more practical. You could use a mains timer switch if needs be.