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 ex-England hooker, 57, won the competition with Devon in 2004 as a player and as coach in 2007, and played in Cornwall's triumphant 1991 campaign.
Dawe has been Cornwall head coach since 2013, leading them to victories in the 2015 and 2016 Twickenham finals.
"It's a nice local game for me, but I'm travelling as a Cornishman," he said.
"I'm 100% behind Cornwall, there's no emotion at all."
The game will be played at Ivybridge on Saturday, with Cornwall favourites as they continue their pursuit of a hat-trick of County Championship titles against a Devon side who were only promoted to Division One because of a restructuring of the competition.
Devon head coach Dan Parkes played under Dawe at Plymouth Albion, and acknowledges his side are underdogs.
"It's going to be hard, the boys are going to be up against it," he told BBC Sport.
"For Graham the pressure is on, but I'm sure they'll get to their third final.
"The better we can do, the more likely we can attract players next year. Players play for Cornwall because they know they have a great chance of reaching Twickenham."
The way arts bodies responded publicly to a critical review was "not helpful", Economy Secretary Ken Skates has said.
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Graham Dawe says he has "no emotion" ahead of his Cornwall side's County Championship game at Devon, who he previously played for and coached.