Gainsborough was the nearest registry office to Broadholme. Her mother was married before.
Oh I see. However my great grandmother was also born Broadholme and she was registered Newark
Historically, Broadholme was in Nottinghamshire, and came under the Newark Registration District, and a birth in Broadholme would have had to be registered in the Newark District. Distance doesn't come into it - it's all about which district it happened in.
There are a few nearby places, eg Kettlethorpe and Newton on Trent, which were in Lincolnshire and in the Gainsborough Registration District. So if she was born while her mother was visiting one of these places, they'd have had to register her in the Gainsborough District.
(In 1989 Broadholme was transferred to Lincolnshire and the Lincoln Registration District, as it then was, but that's not relevant to the question here.)