I was actually trying to find some information about "My Daddy" a few days ago, as I'd never heard of the song and therefore found it strange that it would be charting. I very nearly concluded it didn't exist, until I came across the following record by the London Piano-Accordeon Band: (https://www.45worlds.com/78rpm/record/g24277), which is titled "My Daddy" and has composer credits to Box, Cox and Pelosi (presumably Don Pelosi). There was a corresponding UK release (RZ MR-3251) that was mentioned in The Gramophone April 1940 issue, which is consistent with the Phyllis Robins version apparently charting in May.

I couldn't find any other versions though, and I still find it strange that Phyllis Robins would have recorded such an obscure tune - her other records seem to suggest she confined herself to the big hits of the day. I do wonder if the Missing Charts book has got it wrong, and the title should be "My Heart Belongs To Daddy", although then again May would be a little late for a version of that tune.