The responsibility for dealing with black mould very much depends on the cause. The landlord is responsible for mould if it’s caused by leaking pipes, structural defects, or an inherent damp issue. This would usually include mould caused by rising damp and penetrating damp.
But the tenant could be responsible if the mould is caused by condensation because of their lifestyle; for example, if they’ve been drying clothes on radiators and not opening the window. You can find out more here.
Mould can often be caused by an ineffective fan, or the tenant turning the fan off. It is critical that landlords check fans in wet areas (kitchens and bathrooms) to make sure that they are operating correctly.