Thinking about pronunciation in a thick accent I wonder about this birth with no MMN
MONAGHAN, MARY ELLEN -
GRO Reference: 1901 S Quarter in GATESHEAD Volume 10A Page 1102
Added… forget that. Looks as though she died 1904
Playing devil's advocate, I think the Monaghan birth in Q3 1901 may still be a contender.
there was also a Mary Ellen Monaghan, registered in Gateshead in Q4 1901, MMN McCormick.
Marriages Q4 1900 John Monaghan and Catherine McCormick
and the child who died in 1904 was buried at Gateshead East on 9th Feb 1904, age said to be 2 and was the daughter of John Monaghan.
https://www.familysearch.org/ark:/61903/3:1:3Q9M-CSJ7-H9G8-8?emmadog, it may be worth a shot to email Gateshead Registrars , explain that you have been unable to source a birth reg with the surname Mallachin and that the Mary Ellen Monaghan registered in Q3 Gateshead is an outside chance possible.
Ask if they have time to check the register and be able to confirm/deny whether the mother was Margaret (no father listed) and the child was born 1st Sept.
Way back pre- pandemic Gateshead Registrars were willing to answer email queries like that, they may not have the staff/time atm but its worth an ask? - shy bairns get nowt :-)
Boo