In the 1938 Directory, under the heading Parks and Recreation, section, Gardens. 3 gardens are mentioned, Hamilton Sq, Pearson Rd, Tranmere, Warrington St, Higher Tranmere.

Hamilton Sq, no need of mention.
Warrington St Gardens are still there.

Pearson Rd Gardens?

The images below, directory, map 1888-1913, present day,1911 directory, 1950s.

It appears the houses on the left coming up from Argyle St South were built between 1902 & 1906, by 1911 they numbered 1 to 41, so the waste ground that's present on the modern day map couldn't have been the gardens in 1938, my initial thought.

The land between Holt Hill and Hinderton Road which starts behind Pearson Road may be the gardens.

I have a feeling this was converted to allotments type land in WW2 and it wasn't reverted. One plot between Holt Hill and Hinderton at the western end was up for Sale not long ago. Most of the house gardens look like they have been extended.

It seems very likely, DD.

Actually it was allotments before WW2: 1924