The adverb "quickly" describes the manner in which the action of cutting hair is performed. It modifies the verb "cut".

The other options are incorrect because:

• "Quick" is an adjective, not an adverb.
• "More quickly" is grammatically incorrect because the adverb "quickly" is already in its superlative form.
• "Quicker" is an informal and non-standard form of the adverb "quickly".

Therefore, the complete sentence is:

"He usually cut my hair quickly."