If I own a construction business and a customer has not paid the amount due that it states in our contract, what action can I take?

You don't have to take the customer's claims of poverty at face value--or even if they are true, decline to take action due to them.

You can and should sue the customer if he won't pay; for smaller amounts, sue in small claims court, representing yourself, while for larger amounts, you would hire a lawyer. You will sue on the basis of breach of contract and also unjust enrichment (he received value, the work done, without paying for it).

Assuming you win, you will get a judgment in your favor. If he still does not voluntarily pay, a number of different mechanisms will be available to you to collect, such as putting a lien on real estate; leving on (taking money out of) a bank account; executing on (forcing the sale of) personal property, like a boat or car; and/or garnishing wages.