As for the fruit I cannot say - could be a gage; wheather plum or gage
the prunung advice is the same.
First amy pruning on stone fruit should be done when the tree is
growing not in the dormant season. The reason is due to a disease
called silver leaf which enters in winter through the cuts. There is
no cure and the tree will need to be cut down and burnt to stop the
spead of this disease.
Pruning plum trees is typically a light opperation. It involves the
usual removal of dead, diseased, dieing, crossing, weak spindly
unproductive growth. Crowded heads cn be thinned out. If the cariety
is a spreader like Victoria plum, then cuts are made to an upward
growing shoot. If the new growth is extensive and too much, it can be
reduced buy 1/3rd to a half.
Generally undertake plum pruning after picking. Clearly your tree
isn't in its formative years, for which pruning is started in spring
after bud burst.
Clifford
Bawtry, Doncaster, South Yorkshire