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It's unlikely your GP would send for a scan, however NICE offer the following guidance to clinicians...
https://cks.nice.org.uk/topics/long-term-effects-of-coronavirus-long-covid/
Granted it is likely CoVID related but there are other causes of headaches which should be excluded. To be fair most headache variants are a clinical diagnosis and don't need a scan, but they should at least have some interaction with you.
If they don't find anything ask about long CoVID clinic referral. GL
Late response but I’ve had long covid abt 2 yrs now and all that time had a persistent headache every day along with the other more reported symptoms of fatigue brain fog etc. I’ve been told it goes with the condition and can’t have my noggin scanned to check nothing more sinister, and not managed to convince anyone that it might be worth checking just in case😂
I think it’s likely to be another part of long covid and this seems to be the received wisdom, in both our cases do feel it would be worth checking like those ads on telly tell you to, should ask ur doc for scan to be on safe side, wishing you luck with that