Pang- is used to mean "a tool for (action of the root), used by (root), used for (root), used during (root) serves the purpose of (root), for the purpose of (root)"
The "-ng" sound at the end of the prefix changes to
- "m" before roots that begin with P and B | Example: pambukas (thing used to open, like a can opener)
- "n" before roots that begin with D, L, S | Example: panlangoy (used during swimming, as in a swim suits)
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In your examples, "Sino ang pagkain yan?" translates to "Who is that food?" which is unclear. I am assuming it should be "Para kanino ang pagkaing 'yan?" which is "Who is that food for?"
The answer should be "para sa bata" ("for the kid").
Pambata could be used like this:
"Pambata ang damit na 'yan." - Those clothes are for [the use of] kids.
In the second example, "pang-umaga" means that the job is for the morning.
Hope this helps.