Often, when I have some tradesperson come to the home (eg. handyman), or go to a store here in " ", the person asks that I write out my payment check to "cash". I am aware that this is often to avoid having to declare this payment as income. Can that be justified by halacha?

Though your conscience is right to think as you do, you are not committing an aveira. Rav Lior Shlit"a (Chief Rabbi of Kiryat Arba) wrote in the Hebrew counterpart of this website that the obligation to pay taxes falls on the person receiving it is not considered facilitating an aveiraמסייע לעובר עבירה) ) if you would write him "cash" . Your unwillingness to do so is not an obligation according to Rav Lior, but it is an act of piety. However the handyman is required to pay his taxes because of "dina d'malchuta" and if he does not pay he is considered a thief.