"Dominic Raab, the Brexit Secretary, joined a number of ministers in thanking the UK's negotating teams for their hard work".
Mrs May then travels to Brussels on Wednesday to address a dinner for the other 27 European Union leaders.
The prime minister said she was "convinced" that if the cabinet retains a "firm" unitary position that Britain would get a breakthrough in negotiations.
"We are trying to get the message across, not only to Theresa May, but a lot wider, that what we want is a deal that is sensible, that works for Northern Ireland, the whole of the United Kingdom and as I said in Dublin, works for our nearest neighbours".
In his letter, Tusk said: "As you remember from Salzburg, we wished for maximum progress and results that would lead to a deal in October".
"She would have said "look if the Irish want to shoot each other they will shoot each other whether there's a hard border or whether there is a soft border, that is something the Irish will do if they want to".
According to EU's chief negotiator, Michel Barnier, the backstop would only apply to Northern Ireland.
The border between Northern Ireland the Republic of Ireland remains biggest sticking point in negotiations.
Both sides are now focusing on another summit in November, with the hope that differences can be squared at a further meeting before Christmas.
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The EU has proposed a so-called backstop solution that would keep Northern Ireland in the EU's customs union and single market for goods, a solution May has rejected, whose minority government is supported by the Northern Irish Democratic Unionist Party (DUP).
Speaking on the eve of the summit in Luxembourg, Barnier said many terms of the divorce have been agreed upon already, such as how much money Britain owes the EU. He said: 'We have to give [Mrs May] room to negotiate.' He also delivered a specific warning to his predecessor Mr Johnson, saying: 'Boris is a backbencher so... he is free to say exactly as he thinks.
The mammoth three-hour meeting came a day after eight Brexit-supporting ministers took the unusual step of meeting over pizzas in the office of Commons Leader Andrea Leadsom to discuss their concerns about the PM's stance.
The representatives of various European Union governments say the bloc is prepared for the possibility that the drama could conclude in a bitter dispute and with no agreement at all - even after three, four or five acts are tacked on to the production.
Irish deputy prime minister Simon Coveney said earlier on Tuesday: "I never expected agreement to happen this week".
"We will take this time, calmly and seriously, to find this global agreement in the next weeks", he said.
The unique circumstances in Northern Ireland and the timeframe of the Brexit implementation period create the need for a backstop, May argued.
Although this is in no small part thanks to Brexit and the attendant calls for a vote on Irish unity, it is at least notionally over issues, like the status of the Irish language in Northern Ireland, that could have been resolved long ago if the DUP wasn't so unyielding.