Greek prince and daughter of Zeus I see around lake, loving Greek culture (12)

I believe the answer is:

philhellenic

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'greek prince and daughter of zeus i see around lake loving greek culture' is the definition.
I can't judge whether this defines the answer.

'greek prince and daughter of zeus i see around lake' is the wordplay.
I cannot really understand how this works, but
'greek' could be 'hellenic' (synonyms) and 'hellenic' is found within the answer.
'prince' could be 'p' (abbreviation) and 'p' is found within the answer.
'i' is present in the answer.
'lake' could be 'l' (geographical abbreviation) and 'l' is present in the answer.
A single letter 'h' remains which might be clued in a way I don't see.
This explanation may well be incorrect...

Can you help me to learn more?

I've seen this clue in The Independent.

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