The MP, who was born in the eastern European country, formally requested that its government veto any move for a Article 50 extension that would delay the UK exiting the European Union.
He tweeted on Tuesday: "Lobbying Polish parliament to support Britain in concerns over Northern Ireland backstop.
"Our friends the Poles have taken a lead in EU to better understand our perspectives and help us break impasse with Ireland and EU.
"Asking our Polish allies to block any attempts by EU to permit extension of Article 50. For everyone concerned, we leave at 11pm on March 29 as planned."
Mr Kawczynski's tweet was shared widely online and lampooned by many remain-supporting journalists and MPs such as former cabinet minister David Lammy, who described the request as "extraordinarily reckless".
Tooting Labour MP Dr Rosena Allin-Khan also shared the tweet and said: "Nothing says 'protecting sovereignty' like asking a foreign Government to veto decisions being taken by our Parliament."
And now Dr Laura Davies, Labour's parliamentary candidate for Shrewsbury and Atcham, has criticised Mr Kawczynski and called for transparency.
She said: "I think it’s appalling that a member of parliament should try to use his personal influence to undermine the sovereignty of parliament. I have asked him to publish the details of his approach.
"Constituents here in Shrewsbury and Atcham deserve this transparency.
“There is no majority in parliament for crashing out of the EU without a deal. A no deal Brexit would be a disaster for jobs and the economy.
"It is simply not a viable option. Instead of working towards a solution, he is in fact trying to prevent parliament from reaching it."
Mr Kawczynski's letter comes as a cross-party group of remainer MPs tabled amendments that aim to block a no deal Brexit and potentially delay Brexit for up to two years.
“Several MPs from other parties have tweeted their condemnation of his actions," Dr Davies added.
"This comes just days after he mistakenly tweeted an interview in which the Polish government described a no-deal Brexit as a catastrophe.
"The Tories have no coherent or achievable plan for Brexit and now a Tory backbench MP is trying to subvert the sovereignty of parliament. He should seriously consider his position.”
Mr Kawczynski has been contacted for comment.