Yes, this is process is called a ‘transfer of equity’ and you can add another party subject to criteria and status.
To do this, you’ll need to apply to the lender first and work with a solicitor to process the legal work and transfer the property into the new names/remove a name.
If you're adding or removing a party as part of your remortgage, you can still qualify for free standard legal work if a transfer of equity is done as part of the remortgage process with TSB. But you’ll need to pay an extra fee for this service. The cost for a transfer of equity is typically around £300 payable to your solicitor.