The desire to break a construction contract can arise for several reasons. The consumer can want to withhold payment if work is performed poorly or behind schedule. The contractor might want to be freed from the contract if it seems the compensation is too small for the work to be performed. Construction contracts, like any contract, are legally binding, and can therefore create significant liability if they are broken outright and unilaterally. Ideally, one can shed the obligations under the contract without committing a legal breach.
Renegotiate. Contracts are important to a free market society because they're intended to be wholly determined by the parties entering into them. If either party to a contract finds the terms are unacceptable, the first and best option is to renegotiate for a new contract. In a construction context, this can mean paying for completed work that is acceptable and writing a new contract for the work that remains.
Invalidate the contract. If one party strays far from the terms of the original contract, it can invalidate the entire agreement. For example, if you sign a contract for someone to build a fence around your yard, but they install a pool, you would probably not be obligated to pay them under the contract. A contract might also be invalid if there is evidence of fraud, such as if the contractor stated that he had a license or insurance, but really did not. Conversely, the contractor would probably not be responsible for a contract if the other party was not the owner of the property and legally capable of entering a contract.
Check for breach by the other party. If renegotiation is not an option, and the contract appears valid, you might get out of your responsibilities if the other party has breached the agreement. A minor deviation from the terms of the contract that does not produce damages will probably not affect the agreement. A material breach can give you a claim for compensation in the amount necessary to correct the deviation, and cause for a lawsuit. In the alternative, a material breach can be used as leverage to cancel or renegotiate the contract.