" "The" is optional - the sentence makes good sense with or without it.
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Vincent TeoCan I say,No. The expression is 'give sth to sb" or "give sb sth". You must use 'to', not 'on'. I would use 'to make/help them grow', not 'let them grow' or 'for growing'.
(a) She gives (the) fertilizer on the plants to let them grow.
(b)She gives (the) fertilizer on the plants for growing.