Ommiting short-term position and background check

You mentioned you're from the UK so my answer is based on that:

Your CV is about showing your relivent work experience, not about giving them a play-by-play of everything you've done over the past 10 years. Missing out a 1-2 month position is not a big deal and nothing to worry about.

Be prepared as they may ask your for a reference for this job, but (as you mentioned) you've already declared it on the back ground check application, which is exactly what that application is for!

My question is whether the employer can revoke my offer once he realizes I had a short-term position that I didn't mention?

You didn't mention a country, or if this was an official offer etc... but in general:

It sounds like you haven't actually (officially) been offered the job yet. So yes, the employer probably could turn around and decide not to employ you, and can do this for any other reason so long as it's not discrimination.

When I was completing the background check application I did put details of this position so the gap can be explained.

Is it a CV or resume. They have different meanings, or the same depending on country. Usually on a CV you would be expected to include it, but not on a resume.

It sounds like you provided a resume, and as you provided details on it when you completed the background check application, you have provided all the detail required.

So personally I don't think you need to contact HR and explain it however, if you want to, I don't think it will cause any harm.