basement is a second home

[Ferrytoll]

Bought a house unaware that the basement classified as a second home. Us, a family of 6 living in the house. Council demanding the second home property tax.
If the housewife would register as living in the address of a basement flat would she be able to claim Council tax relief as she has no income?

[PallasAthena]

Is the basement classed as a separate property by the council for Council Tax? If so, why? Perhaps it was separately let out by the previous owner?

If you and your family are now occupying the whole house - including the basement - as a single residence then the basement should not be separately rated for Council Tax and you need to notify the council of that. The whole house should be rated as a single property. If the basement is not physically connected to the rest of the house you might need to make a connection.

If the housewife would register as living in the address of a basement flat would she be able to claim Council tax relief as she has no income?

I don't understand what you mean but Council Tax liability is not linked to income so having low or no income does not exempt you from paying CT. Councils may offer CT financial support to people on benefits but that is not relevant to the basement if you are actually occupying the whole house as your main or only residence.