も構わず【もかまわず】

も構わず means "without caring about", "regardless of": someone goes ahead and does something while ignoring a thing that would normally stop them or make them hold back. It follows a noun directly; a verb or an い-adjective takes の first.

Without the の the verb has nothing to attach to: ×汚れるも構わず, ×痛いも構わず.

The thing ignored is usually other people – ひとめ人目, まわり周り, めいわく迷惑 – or a danger or discomfort. The sentence describes someone else's behaviour from the outside, and often disapproves of it.

も構わず or にもかかわらず

にもかかわらず states a plain contrast between two facts: X is true, and Y happened anyway. も構わず is about the person's attitude – they knew and did not care.

The match is not a person and has no attitude, so ×雨も構わず、試合は行われた does not work.

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も構わず belongs to writing and narration, and is often spelt もかまわず. In conversation you would say きにしないで気にしないで or き気にせず.