Evicting a tenant advice please

Hi , new here
Been directed to here , by someone on another ( unrelated ) forum and hope you guys can help.

A relative of mine have issued a section 21-4a notice on their tenant . The tenant with a week to go have said they havemnt got enough time to get out - not sure if its because the rent has been pretty low and they are struggling to find a similar house at the same price , or whether they are just "Playing a game" ?

Obviously , if it comes to it , they will issue court procedings to get them out , but would prefer to sort it "amicably" if possible

My question is
If they said to the tenants ," well if you give us £x rent , then we will let you stay for another fortnight ( say) ", and they agree , will actually having this "side agreement" ( actually just getting just the equivalent of 2 weeks rent) , make the section 21-4a notice null and void because they have let the tenants stay on after the expiry date with approval ?
And hence would they then have to do another 21-4a and give 2 further months notice ?

Hope that makes sense ? ( new to all this )
Thanks