Can I own a pet while renting?

Can I get a pet in my current rented home? And, how can I persuade a new landlord to rent to me with a pet?
My home has a mould problem
No matter what I do I can’t stop condensation, avoid damp and prevent mould in my rented home
My landlord is not fixing something
I told my landlord about a leak/broken boiler/mould outbreak/infestation and they won’t do anything about it.