As a commenter, I don´t like the sentences at all. Literally,
Kann ich noch einen Apfel?
would mean something like
Can I another apple? (no "have" etc. here)
People will understand that you want to have another one, but still...
Append a essen/haben/nehmen/kaufen/... (eat/have/take/buy), whatever you want to do. And even better than
Kann ich noch einen Apfel nehmen? <=> Can I take another apple?
Darf ich noch einen Apfel nehmen? <=> May I take take another apple?
It´s nitpicking, but you don´t need t ask the other person if you are able to take it.
About noch, einen and mehr: Let´s start with a simple
Darf ich einen Apfel nehmen?
May I take a apple?
May I take one apple?
Ok when you´re asking for the first apple, strange if you have already one.
Darf ich noch einen Apfel nehmen?
May I take another apple?
You have already one or more apples, but you still want another one.
From another angle: Repeated "one". Asking multiple times with einen would imply that you´ve forgotten the apples you already have, so starting with the second question, add a noch
(and don´t ask 10 times for a single apple)
Darf ich mehrere Äpfel nehmen? (attention, plural Ä)
May I take multiple apples?
You currently have no apples and want an unspecified amount (at least two, else use einen).
Instead of multiple, you can/should insert an specific amount.
Darf ich noch mehr Äpfel nehmen? (attention, not noch mehrere)
May I take even more apples?
You already have some apples and want more.
Darf ich mehr Äpfel nehmen?
May I take more apples [than I already took / than you just gave me /
than my brother took]?
Depending on the situation:
a) It´s similar to noch mehr (you have already and want more), but not suited for repeated questions (at least starting with the second mehr, add a noch).
b) You want more than some comparison amount (brothers apples etc.). In this case, it doesn´t matter if you have already some apples or not.