The coronavirus response in Wales risks "raising questions about the value of devolution", Welsh Labour members have said.
Welsh Labour Grassroots criticised ministers in Cardiff for being "largely in lock-step with the UK Government".
The group, on the party's left, raised concerns around the supply of tests and personal protective equipment (PPE).
First Minister Mark Drakeford said in response he agreed it was important Wales had its own approach.
Welsh Labour Grassroots is the sister organisation of Momentum, the group set up to support former Labour party leader Jeremy Corbyn.
In a letter to the first minister, the group raised specific concerns around testing, protective equipment for health workers and support for renters.
But, while members of Welsh Labour Grassroots "applaud the efforts" of the Welsh Government during the crisis, they also raised concerns the "approach to policy-making appears at times to be indistinguishable from that of Downing Street".
The letter, sent on 5 April, added: "We fear that if Wales does not develop a response that goes above and beyond the UK government's strategy, we not only risk raising questions about the value of devolution and a Welsh Labour government, but risk the well-being and lives of potentially tens of thousands of people.
"The Welsh Government will quite rightly be held to account for the way in which it deals with the COVID-19 crisis in Wales," they said.
The group of party members raised particular concerns around the supply of coronavirus tests and PPE for healthcare workers, as well as a lack of protection for tenants in private rented accommodation.
In the letter, they called on the Welsh Government to:
- Put testing at the very top of its coronavirus response
- Put the manufacture of PPE "on a wartime-like footing" to ensure a consistent supply
- Suspend council tax for the most vulnerable
- Clarify whether it has the power to implement an immediate rent freeze
Mr Drakeford, who enjoyed the support of Welsh Labour Grassroots during the Welsh Labour leadership campaign of 2018, said in response he agreed "it is very important we have our own, distinctive approach to dealing with coronavirus in Wales".
He added that an England-focused media that "struggles to understand devolution" made it harder for his government to communicate some of the different approaches being taken in Wales.
On PPE, he said: "All four UK governments are agreed that our ability to secure PPE in the face of global shortages depends on us combining our purchasing power and sharing our procurement capacity.
"We are supplementing the draw down we are able to make from those UK supplies through additional contributions from Welsh industry.
"PPE has to be provided to strict and specific standards. It is not the case that every offer of help, even when very genuinely made, can be turned to use."
But the Wales Trades Union Congress has told BBC Wales they are seeing "horror stories" over a lack of PPE and has called for greater clarity on the supply of kit.
The first minister said he was following the testing regime supported by Wales' Chief Medical Officer, Frank Atherton.
"If his advice changes, our actions will change accordingly," he added.