Looking for a Charles Garner born 1890 living in Staffordshire at the 1939 register the one mentioned in post #6 seems the only suitable person.
32 Grosvenor St
Charles Garner, 30 Sept 1890, plumber & painter
Harriet Garner, 22 Feb 1891
Charles Garner married Harriet Stuckey Q2 1912 Stuckey, they had children in Stoke on Trent :-
Evelyn, b 21 Dec 1912, m Harry Boone 1941, died 1945
Frances, b 1920, d 1922
Maude E, b 1919, married William Lyons Q3 1940
you say your grandfather Frederick was eventually brought up by his father Charles, could he have been taken in to the above family? If he was born illegitimate in 1911 and then his father married another woman in 1912 the father, Charles, would then be in a position to offer him a home. If the mother was struggling to keep him on her own or if she wanted to marry but her husband would not accept her illegitimate son then in order to take him on Charles must have had a wife, in those days a man was very unlikely to bring up a child totally alone.
Do you think that the above Evelyn/Frances/Maude could have been his half sisters? Maude & William Lyons had children Evelyn A Lyons, 1945, and Denis Lyons, 1947. They may possibly know more about the situation. It may be possible to contact them at addresses on the current electoral register, please send me a PM if you would like help with that.
Good luck!
Elaine