Ap Iorwerth’s comments come after he hosted the nationalist first ministers of Scotland and Northern Ireland in Cardiff last month and jointly declared that “Westminster’s time is coming to an end”.
In his 2,000-word letter to Burnham, ap Iorwerth makes a detailed case for the transfer of powers over policing and justice, rail infrastructure, and the Crown Estate, external to Wales.
He also calls for “funding arrangements that are fair, needs-based and reflect Wales’ demographic and economic realities”.
He adds that “Wales’ constitutional settlement should continue to evolve to improve democratic accountability, boost growth, strengthen public services and deliver better outcomes for our communities”.
Ap Iorwerth is expected to meet the prime minister, along with the first ministers of Scotland and Northern Ireland, at the end of the month.
Andy Burnham has previously said he is “open” to the idea of giving the Welsh government control over policing.
He has also said: “I have to respect the governments in the nations of the UK, and I will, and give a fair hearing to anything that is put to me.”