meaning of 'to give someone gloves'?

I was not engaged to Mildred, but I lodged with her mother, and I sang duets with Mildred and gave her gloves when it would run to it, which was seldom. She was a dear, good girl, and I meant to marry her some day.

This means that the narrator had an emotional attachment to Mildred, and expressed it by giving her an occasional gift. They would be taken as an expression of esteem, without crossing a boundary in an uncomfortable way. The narrator wasn't ready to give Mildred a home, or even a ring, but he hoped to some day.

In that cultural context, women wore kid gloves when they went out of the house. Gloves would have been a substantial gift but not extravagant. Kid gloves wore out and needed to be replaced regularly. I'm guessing the frequency would have been approximately the same as people replace running shoes nowadays -- on the order of pproximately every two years.

My German mother-in-law, born in the 1920's has a pair. They are a work of art.