Throwing lines on river catching a fish (7)

I believe the answer is:

roaches

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'throwing lines' is the definition.
I can't judge whether this defines the answer.

'on river catching a fish' is the wordplay.
'on' means one lot of letters go next to another.
'river catching' becomes 'es' (I can't explain this - if you can you should believe this answer much more).
'a fish' becomes 'roach' (roach is a kind of fish).
'es' put after 'roach' is 'ROACHES'.

Can you help me to learn more?

I've seen this clue in The Guardian.

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