On 21 March 2011, Cleaver made his first-class debut for Central Districts in a rain-affected draw, where only 8.2 overs were possible in the match.
[4] In February 2020, in the 2019–20 Plunket Shield season, Cleaver scored a double century in the first innings and made nine dismissals.
[9] In June 2022, Cleaver was added to New Zealand's Test squad, for their third and final match against England,[10] but did not play.
Later that same month, he was named in New Zealand's One Day International (ODI) and T20I squads for their tours of Ireland, Scotland and the Netherlands.
[11] Cleaver made his T20I debut on 18 July 2022, for New Zealand against Ireland,[12] and in the second match of the series he hit a match-winning 78 not out.