If you've reported a problem to your local council and they've inspected the property under the HHSRS, an improvement notice is one of the enforcement tools the council can use. It's the most common response to hazards that need fixing but aren't immediately dangerous.
The notice specifies exactly what works the landlord must carry out and sets a deadline for completion (usually 28 days or longer depending on the complexity of the works). The landlord has a right to appeal the notice at the First-tier Tribunal within 21 days, but if they don't appeal or the appeal fails, they must comply.
If the landlord doesn't carry out the required works, the council can prosecute them, issue a civil penalty of up to Β£30,000, or carry out the works itself and recover the costs from the landlord. Failure to comply with an improvement notice is also a qualifying offence for a rent repayment order, meaning you could claim back up to 24 months' rent.