叩く and 戦う
Etymologically, Shogakukan's 国語【こくご】大辞典【だいじてん】 and 大辞林【だいじりん】 list just the ふ auxiliary derivation. Meanwhile, 大辞泉【だいじせん】 lists both the ふ auxiliary and the 合う possibility as derivations.
Given the not-uncommon pattern of deriving iterative / repetitive / continuative verbs by appending ふ to the 未然形【みぜんけい】, and given the appropriate semantics here (叩く "to hit something" ⇒ 戦う "to be hitting something (repeatedly)" → i.e., "to fight"), the ふ auxiliary derivation seems more likely.
Other verb pairs with similar derivations include:
- 病む【やむ】 "to get sick" ⇒ 病まう【やまう】 "to be sick" (obsolete)
- 語る【かたる】 "to talk, to tell" ⇒ 語らう【かたらう】 "to be talking, to be telling"
- 住む【すむ】 "to settle (in a place)" ⇒ 住まう【すまう】 "to be settled", i.e. "to be living in a place"
The original ふ often just became う, but in many other cases, this underwent a further shift into the 已然形【いぜんけい】, changing from ふ to へる to える. This expands the list of pairs to include examples such as:
- 押す【おす】 "to push, to press" ⇒ 押える【おさえる】 "to be pushing, to be pressing" → also "to repress, to hold down"
- 取る【とる】 "to take" ⇒ 捕らえる【とらえる】 "to be taking, to take and hold", i.e. "to capture"
- 踏む【ふむ】 "to step, to tread" ⇒ 踏まえる【ふまえる】 "to be stepping", i.e. "to be of a position, to have taken a stance"
The key semantic shift is that the bare verb expresses a momentary action or change of state, while the verb + ふ expresses an ongoing or repeated action or state.