Where situations warrant, a good contractor will properly explore the prospective job site in advance of making a quote or a contract.  Often in these cases, the contractor does not look for more money.  However, some contractors, through accident or design, end up looking for more money part way through the project.  Absent some exceptional circumstances, the contract price put forwarded by the contractor would still apply.
Where we most often see requests for more money are from contractors who were not properly vetted and satisfied all the criteria in our checklist.  Often these are small contractor operations who do not have a lot of money on hand and are either poor at quoting for projects or else are not able to execute them well.  In either event, they find themselves short on money and therefore without the means to finish the project.  This puts the homeowner in a difficult position.
If the work is generally of good quality and the additional amount requested is within 10% of the contract price then the homeowner may wish to pay it.  (It should be repeated that under the Consumer Protection Act a contractor may increase their price by up to 10% of a quoted price.)
If, however, very little work has been done and the contractor is looking for the balance of the contract price then this can be harder to determine how to address the situation.  The contractor has already been exposed to have cash flow issues and there is no guarantee that further payments will result in further work.  In fact, we have seen situations where the contractor took additional monies and then disappeared altogether.  One potential way to manage such a request is to agree to pay for any materials or subtrades directly.  For example, if the work involves a delivery of tiles then the homeowner could pay the supplier directly as opposed to paying the contractor.
If, however, the request is excessive and the homeowner will not acquiesce to it then this will could create a situation where the contractor ends up being in breach of the contract.