Penistone and Stocksbridge MP Angela Smith fears her party will be punished at the ballot box by voters in the region and slip behind the Tories into second place if Parliament ultimately votes for Brexit to go ahead.
A Government source speaking to The Times said Bercow allowed the amendment despite Commons clerks, whose job it is to advise the Speaker on Parliamentary procedure, telling him that amendments should not have been allowed.
MPs voted 308-297 on Wednesday in favour of demanding the government come up with an alternative plan within three working days after Tuesday's vote, rather than a planned 21-day limit, in a non-binding motion that nonetheless piles pressure on the government.
The defeat highlights May's weak position as leader of a minority government, a divided party, and a critical parliament just days before she is due to hold a pivotal vote on whether to approve the Brexit deal she has negotiated with the EU. Some investors and major banks believe May's deal will be defeated on Tuesday but that eventually it will be approved.
Informing the House of the move, Brexit Secretary Stephen Barclay accepted that it would not by itself win over the Tory rebels and DUP allies who are threatening to send the Prime Minister's deal to defeat next Tuesday.
Angry Tory Brexiteers questioned the Speaker's impartiality over Brexit in heated exchanges.
But a new survey of MPs published on Wednesday suggested that lawmakers will struggle to unite around an alternative to Mrs May's deal or a "no deal". She claimed that having listened to concerns from MPs, she would seek the necessary changes in order to get her Brexit plan through Parliament.
Barclay told MPs that voting for May's deal would instruct the government to obtain further assurances from the European Union that the Irish backstop "would only be a temporary arrangement".
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Labour said it would try to trigger an election by calling for a no-confidence vote in the government if May's deal is defeated next week.
But without the DUP's support, and with many of May's Conservative MP still strongly opposed, the deal can not pass.
Former Conservative minister George Freeman accused Mr Corbyn of facing two ways at once, behaving like "a Brexiteer up north and a Remainer down south".
Mr Bercow selected a proposal from former minister Dominic Grieve, which attempts to speed up the process for the Government to reveal what it will do next if Theresa May's deal is rejected.
Irish prime minister Leo Varadkar has confirmed that the EU is "happy" to give assurances that they "don't want to trap the United Kingdom into anything" in response to claims the Irish border backstop could be used to keep the United Kingdom in the customs union.
He wrote: "Parliament can establish that it wants no deal to be ruled out".
May is also seeking assurances on the operation of the backstop from European leaders, which she hopes to deliver before the vote next week, although they say they will not reopen the deal. The vote, which saw 20 legislators from May's Conservative Party rebel and side with the opposition, indicates that a majority in Parliament opposes leaving the European Union without an agreement and will try to stop it happening.
Labour took 49 per cent of the vote in Yorkshire and the Humber compared with the Conservatives' 41 per cent in last year's General Election, but an opinion poll of 2,000 people in the region conducted by YouGov for the People's Vote campaign says this has already fallen to 44 per cent.
In the Commons, Tory anger was directed towards Mr Bercow who many MPs have long suspected is unduly sympathetic to the Remain cause.