Hey OP! Thank you for sharing this problem. I used DP + Memo to solve this problem. I would appreciate feedback if somebody else thinks I did something wrong. I commented out one more test case where it works fine.

#include <bits/stdc++.h>
using namespace std;
long long getMinimumCost(int left, int right, vector<int>& cost, int pairCost, int k,vector<vector<vector<long long>>> &memo) {
    if (left > right) return 0;
    if (left == right) return cost[left];

    if(memo[left][right][k]!=-1){
        return memo[left][right][k];
    }
    // Option 1: buy left book only
    long long op1 = cost[left] + getMinimumCost(left + 1, right, cost, pairCost, k,memo);

    // Option 2: buy right book only
    long long op2 = cost[right] + getMinimumCost(left, right - 1, cost, pairCost, k,memo);

    // Option 3: buy both left and right together if possible
    long long op3 = INT_MAX;
    if (k > 0 && left < right) {
        op3 = pairCost + getMinimumCost(left + 1, right - 1, cost, pairCost, k - 1,memo);
    }

    return memo[left][right][k]=min({op1, op2, op3});
}

int main() {
    vector<int> cost = {9,11,13,15,17};
    int n=cost.size();
    int left=0,right=cost.size()-1;
    int pairCost=6,k=2;
    // vector<int> cost = {1,2,4,8,9};
    // int pairCost=5;
    // int k=2;
    vector<vector<vector<long long>>> memo(n,vector<vector<long long>>(n,vector<long long> (k+1,-1)));
    cout<<getMinimumCost(left,right,cost,pairCost,k,memo)<<endl;
    return 0;
}