In this paper, a simple and elementary method is given for deriving estimates of sums of arithmetic functions in Fq[t]. The method is the function field analogue of a result first proved by Stefan A. Burr in 1973 in the number field case. A novelty of this paper is that we are able to extend Burr's result, in the function field context, and obtain secondary main terms for the appropriate sums involving the divisor functions dr(f ) with an error term that improves the one given by Burr.