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I think the Crimean government may have a big fight on its hands persuading citizens to endorse joining Russia.
Source: BBC News - Ukraine crisis: Crimea vote 'will not be recognised'

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on someone's hands

  1. Used to indicate that someone is responsible for dealing with someone or something:

    he has a difficult job on his hands

Source: Oxford Dictionaries definition of “on someone's hands”

This phrasing is often used in the conditional sense to indicate that if things go a certain way, then the path will be difficult. It's like saying “these difficulties will become your responsibility (or fall to you).”

In this specific instance, the message is essentially:

I think the Crimean government may [be faced with] a big fight [if it tries to] persuad[e] citizens to endorse joining Russia.

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