He was asked about comments made by his shadow defence secretary Nia Griffith, who said "it would have been easier for us" if Mr Corbyn had clarified that he backed the expulsion of 23 Russian diplomats from the UK.
The Labour leader condemned the "appalling" attack but pressed the PM on whether the United Kingdom had supplied traces of the nerve agent used in the attack to Russian Federation for analysis before Wednesday's deadline, as the Kremlin had asked.
Asked by Sky News whether the UK would be safe with Mr Corbyn in No 10, Mr Woodcock said: "I understand why you ask that question, but I think it's important - at a time when the UK has found itself under attack with chemical weapons - that actually we focus on what the UK Government, the UK state can do and that we all as MPs, of whatever party, give support to that where appropriate".
The Labour briefing had close echoes to the Russian government's response to the UK's condemnation.
Media captionCorbyn: "Our response must be decisive and proportionate and based on clear evidence".
Why has this caused a problem?
"And that is why we are fully supporting the expulsion of 23 diplomats", she told the Daily Mirror.
He added: "It is essential we follow the evidence and what the evidence produces".
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When pressed on the issue and his shadow defence secretary's comments, Corbyn said: "The evidence points towards Russian Federation on this, therefore responsibility must be borne by those that made the weapon, those that brought the weapon into the country and those that used the weapon".
"Can I assure the Prime Minister that most of us on these benches fully support the action she is taking", said former Labour cabinet minister Ben Bradshaw.
Corbyn questioned the Prime Minister's hawkish line against Vladimir Putin, and suggested that Tony Blair's "dodgy dossier" on Iraq makes it hard to trust the evidence of Britain's intelligence agencies in this case.
Speaking in the Commons, Mr Corbyn condemned the "appalling" attack but pressed Theresa May on whether the United Kingdom had supplied traces of the nerve agent used in the attack to Russian Federation for analysis before Wednesday's deadline, as the Kremlin had asked.
The Prime Minister's response is proportionate while also garnering the support of the United Nations, Nato, the European Union and those longstanding allies of the UK whose democracy and security has also been compromised by these acts of aggression.
She also distanced herself from Corbyn's spokesman, saying she "can't speak for Seumas Milne", and that he "has to speak for himself".
"Clearly whoever carried out the attack is responsible for what was a completely heinous and reckless act", he said, arguing that the agent could have fallen into a third party's hands. However, also there is a history in relation to weapons of mass destruction and intelligence which is problematic, to put it mildly.
He said there was "a series of possibilities as to who may be responsible", including oligarchs, Mafia elements and other ex-Soviet states.
Jeremy Corbyn has warned Theresa May not to "rush way ahead of the evidence" over the Salisbury poisoning, in what he called the "fevered" atmosphere of Westminster.