In my opinion, in relative clauses the rules on auxilliary usages don't apply. e: I have / she has etc. so, "me" is singular and takes "needs".
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Anonymousbecause here "need" works as a main verb, when we want to use it as a "semi modal auxiliary verb then we should not use "s" ex: He need to wash the car, here "need" works as a modal and also the word "need" is a "semi modal axuilary verb" with modals we can't use any "s" "es" and "ies".He need to wash the car, here. (Is "need" the correct verb?)
Anonymousbecause here "need" works as a main verb, when we want to use it as a "semi modal auxiliary verb then we should not use "s" ex: He need to wash the car, here "need" works as a modal and also the word "need" is a "semi modal axuilary verb" with modals we can't use any "s" "es" and "ies".The sentence of yours that I have underlined is incorrect.