The Welsh first minister has formally called on Andy Burnham to deliver a "new deal" in his first address to a Plaid Cymru annual conference since taking office.

Rhun ap Iorwerth told party members during a keynote speech in Llandudno on Friday that his government had a "clear mandate" for devolving more powers from Westminster.

The first minister, who beat Labour to lead his nationalist party to a first victory in Senedd elections in May, said he had written to Burnham on Thursday to ask the prime minister for "a new deal to redress the historic unfairness of our fiscal and constitutional settlement and to put Wales on an equal footing with Scotland as a bare minimum".

After a standing ovation, ap Iorwerth added: "A new deal – the first of its kind – whereby a Welsh first minister invites the UK government to genuinely treat Wales not as a region of England but as a nation in its own right. A new deal to secure powers … not as an end in themselves but as a means of growing our economy and improving outcomes for everyone."

The 2,000-word letter to Burnham made the case for transferring powers over policing and justice, rail infrastructure and the crown estate.

Ap Iorwerth recently told the Guardian he had been encouraged by warm initial discussions with Burnham – a stark contrast to his interactions with Keir Starmer. However, the Welsh wing of the Labour party in Westminster is generally sceptical of devolving more powers to Cardiff, especially now Labour no longer controls the Senedd.

The first minister said on Friday: "I do believe the new prime minister and I both want to have a constructive and positive relationship. I've looked him in the eyes and said do not undermine Wales as a nation … do not bypass our Senedd, and he looked me in the eyes and promised he would not. I will hold him to that."

Since becoming prime minister in July, Burnham has made much of his support for regional devolution of power and resources in England, but his approach to the existing devolved administrations in Belfast, Cardiff and Edinburgh is still forming.

Plaid Cymru's Welsh election win ended 100 years of Labour dominance in Wales and blocked the momentum of Nigel Farage's Reform UK. Welsh Labour had governed in Wales since devolution began in 1999 but won just nine of the 96 seats in the new Senedd. Support for the party had been ebbing for years, driven by frustration about the management of public services, which have fallen behind the other UK nations on many metrics.

Ap Iorwerth's win also put nationalist parties in government simultaneously in Wales, Scotland and Northern Ireland for the first time, although Plaid Cymru has ruled out an independence referendum during its first term.

The party's last autumn conference in Swansea had been full of excitement as members sensed the tide turning in Welsh politics. Its first as a party of government remained more relaxed and homely than those of rivals with deeper pockets and more seats in Westminster.

The first minister's demands for a "new deal" came a few weeks after a landmark summit of Plaid Cymru, Sinn Féin and Scottish National party leaders during which they affirmed their respective countries' right to seek independence from the UK.

Support for independence in Scotland has hovered between 45% and 50% since the 2014 referendum. Data suggests it is rising in Wales and Northern Ireland, and stands at 32% and 36% respectively.

Burnham is expected to meet the three first ministers at the end of October.

A UK government spokesperson said: "This government is committed to building a stronger and fairer Wales in a stronger and fairer United Kingdom.

"That's why our focus is now on working with the Welsh government to empower local leaders and communities to deliver good growth and jobs in every postcode."