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'
on someone's hands
- Used to indicate that someone is responsible for dealing with someone or something:
he has a difficult job on his 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.