This is a charade - multiple separate wordplay elements are joined together in sequence to build the answer.
Break down the clue into parts: 'INXS' gives you some letters (with X becoming its Roman numeral value), 'provided' is a word meaning 'if', and 'Yates's first' means take the first letter of Yates. Combine all these elements in order.
Answer: INTENSIFY | INXS becomes IN + TEN + S (where X = ten in Roman numerals), then add IF (provided) + Y (Yates's first letter): IN-TEN-S-IF-Y = INTENSIFY