Let q be a prime power, let Fq be the finite field with q elements and let d1,…,dk be positive integers. In this note we explore the number of solutions (z1,…,zk)∈F‾qk of the equationL1(x1)+⋯+Lk(xk)=b, with the restrictions zi∈Fqdi, where each Li(x) is a non zero polynomial of the form ∑j=0miaijxqj∈Fq[x] and b∈F‾q. We characterize the elements b for which the equation above has a solution and, in affirmative case, we determine the exact number of solutions. As an application of our main result, we obtain the cardinality of the sumset∑i=1kFqdi:={α1+⋯+αk|αi∈Fqdi}. Our approach also allows us to solve another interesting problem, regarding the existence and number of elements in Fqn with prescribed traces over intermediate Fq-extensions of Fqn.