Hi Barbara
I really do sympathise, I too had a long standing problem with a birth reg that did not seem to exist, but the child was baptised (with the correct surname or at least a variation that sounded the same)
Again it was a family who had come to the North East from Ireland. They had a whole squad of children and I had got them all bar one. (there's always one!)
After a very long time I found him. the surname was McGarrigall (or any variation you can think of plus a few you wouldn't)
He'd been registered as McGallagher!
Close, but it took me a while to win the coconut:-)
Given the current number of covid infections there are perhaps the Registrars are short staffed at the moment, its shame as my experience of Gateshead Registrars is that they are willing to help, but perhaps are unable to at the moment as they have other, more immediate, duties to carry out.
I'm assuming that this is the baptism record you have at St Joseph's, Rc , Gateshead
https://www.familysearch.org/ark:/61903/1:1:NJFB-8D7Gives a birth date of 10 Sept 1901
As there is no father named (and the priest would have asked - repeatedly and vociferously!) then we can safely assume she was illegitimate. Given that the LDS would have indexed info for both parents, if that was in the record that's another indication that she was illegitimate rather than an error in the transcription.
I looked again at births registered in 1901, Gateshead on Free BMD , specifying a first name of Mary and a surname as M*, limited it to Gateshead and Sep 1901 to Dec 1901 (Q4 would be pushing it to have the birth registered within the limit of six weeks, but it sometimes happened)
I downloaded those results and looked at each in turn (no matter how close or far away from a version of Mallachin the surname was, and no matter what, if anything, was shown as a middle name)
The only entry that comes up as not having a MMN recorded is the
Q3 1901 Mary Ellen MONAGHAN
which is the one you asked the Registrars about.
As the registrars aren't able to help you at the moment, I am afriad that your only option is to take a punt at getting the pdf from the GRO. Not cheap at £7 a pop, but I can't think of any other way.
If you go that route and the mother is Margaret, and hopefully it has an address that looks likely to be a match, then I'd say that on balance its the right record.
Boo