Using the $S$-type iteration process, we introduce a modified proximal point algorithm for approximating a common solution of the minimization problem and fixed point problem in Hadamard spaces. In particular, we establish strong convergence of the proposed algorithm to a common solution of a finite family of the minimization problem and the fixed point problem of two finite families of generalized $k$-strictly pseudononspreading mappings. Numerical example in support of our main result is given to 14illustrate its applicability. Our work improves and extends some recent results existing in the current literature.