While I can't provide absolute proof, my belief is I am in no doubt that the SH referred to Somerset House which was at the time the site of the General Register Office responsible for issuing birth certificates etc.
It is highly likely that the third element was the individual clerk's invention indicating that a version of the birth cert had been seen. As Gaffy has posted, no place of birth was recorded on a certificate of registration but it would serve as existence of the child. The example S H Cte would seem to be the most sensible. While the certificate of registration was issued locally, it was under the authority of Somerset House wherever it was issued.
As to putting it in the place of birth column, there wasn't another column to put it in and it does at least indicate that evidence of the birth had been seen.
MaxD