In [9], Cetin et al. defined a new special finite sum which is denoted by $C_1(h,k)$. In this paper, with the help of two-term polynomial relation, we will give the explicit values of the sum $C_1(h,k)$. We will see that for the odd values of $h$ and $k$, this sum only depends on one variable. After that we will give many properties of this sum and connections with other well-known finite sums such as the Dedekind sums, the Hardy sums and the Simsek sums $Y(h,k)$. By using the Fibonacci numbers and two-term polynomial relation, we will also give some relations for these sums.