A report for ministers called for Literature Wales to have much of its funding and responsibilities removed.
Literature Wales then described the review as a "dud", filled with "inaccuracies", which its funder, the Arts Council of Wales, also criticised.
Mr Skates said he had been surprised to see "personal statements" that could undermine his response to the report.
The review by Prof Medwin Hughes found the board of Literature Wales was "lacking the skills and experience" to spend public money.
It said there was no evidence of strong governance at Literature Wales and its mission and goals were "unrealistic and vague".
In response, Mr Skates announced that several responsibilities would transfer from Literature Wales to the Welsh Books Council.
They include the Wales Book of the Year award, bursaries for writers and literary events.
Mr Skates has now written to the Arts Council of Wales and Literature Wales to tell them he notes the "concern" they have with elements of the report.
He has asked Prof Hughes and his panel to consider their criticism of the report and to respond during the summer break.
In the letter, Mr Skates assured the organisations that their "concerns are being looked at" but criticised their public responses.
"Some of you clearly perceive serious issues with the report," he wrote.
"You have raised these with me and have agreed with me that the review panel should consider your concerns and respond to them.
"I have therefore been surprised in recent days to see personal statements appear online which could be seen as seeking to undermine the process we have all committed to, without awaiting the panel's response.
"This is not helpful."
In the letter, Mr Skates adds: "I hope and expect that we will move forward in a way that allows this work to be completed in good faith and in doing so maintain productive and positive working relationships, both now and in the future."
The way arts bodies responded publicly to a critical review was "not helpful", Economy Secretary Ken Skates has said.