Q1) They have a right to refuse, if that is what you mean. It depends on how valuable you are to them / how difficult to replace / how important the nature of the work.
Q2) Probably. In law, you have no right to claim unfair dismissal, if that is what you are asking.
You're going to have the bullet, explain the situation and ask them. Personally I would approach the end client (the Water Company) people first, not the agency. Agencies don't normally give a monkey's who'se in there, provided they are getting a propertion of the deal - but it depends on the relationship that you have developed with them.