RIght. You can relax a bit.
If you originally occupied the property under an assured shorthold tenancy agreement and the term has now expired then the tenancy is a statutory periodic tenancy. They must serve a s21 Housing Act 1988 notice upon you requiring possession. The notice period they must give is two months if you pay your rent monthly and it must also expire on the final day of a rent period.
The result of this is that if they claim the letter they have sent you giving 6 weeks notice is a s21 notice and attempt to make an application for an order for possession using the letter then the Court will not make an order for possession. They will require a further notice to be served, giving you the correct notice, with such (two months) notice to expire at the end of another rent period.
If you need to buy yourself time then I would simply not engage with the agent or the landlord until their stated time for eviction is require. A week or so before you should write to them (preferably) by email stating that as the letter only gave 6 weeks notice it does not comply with the provision of s21 Housing Act 1988 and is therefore invalid.
They will reserve the notice giving the correct amount. If the notice at this time is valid and you do not leave upon expiry then they will be forced to apply for an order for possession. A hearing will take place, where you can attend. If the judge makes an order it will likely be the case that you are given a further 28 days or so to vacate.
On the facts, it does not sound as if the agents know what they are doing.
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