could this mean "one more apple" or "another apple"?
No. Unhe is present here. You have to translate that.
"One more apple" could be "ek aur seb (dena)".
The 'और' used here translates to 'and'. The word 'ओर' should be used here in place of और
No the word "ओर" means "towards" or to show a particular direction. For example-
They ran towards east direction - वे पूर्व दिशा की ओर भागे।
But the word "और" can be used for both purposes "and" or "to demand for more".
It seems to me that 'Give them more apple' should also be correct, as apple can be non h count as well (i.e. when apple is presented in pieces).
Or do I miss something?
Would "Give them another apple," also work here? If not, why not?
"Give them another apple" have more than one meanings in hindi as the word another is used for many purposes like "one more" and "different than first" like this
उन्हें एक और सेब दो।
उन्हें कोई दूसरा सेब दो।