The usual reason for being a bit vague about location was that it was the workhouse infirmary. This doesn't necessarily mean the mother was an inmate of the workhouse itself, as its infirmary was the nearest you got at that time to a public hospital. Normally, they just put a street name and number, but maybe there wasn't a number at Saint James Green.
When workhouses closed in the 1930s, many of their buildings were used for similar public assistance purposes, including hospitals and nursing homes, and some occupy the same sites to this day.
