Jackdaw...I was not aware that "dole money" was paid after the State Pension was awarded. Not sure what you mean by "dole money" ?
Is this Job Seekers Allowance, which used to be called Unemployment Benefit, which is what I think of the "dole" ?
Anyway, my central point still stands. Income Tax is only paid on total income, from any source, goes over the £10,000 annual allowance. If the State Pension tips the total annual income over the £10,000, then of course income tax is payable. Nowhere have I suggested otherwise.
By the way, Housing Benefit, along with a 18 other state benefits is NOT taxable. If in doubt, this link will be of interest to you :::
https://www.gov.uk/income-tax/taxfree-and-taxable-state-benefits