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THIS DOES NOT WORK FOR SOME VALUES.. IF YOU HAVE A CLUE PLEASE HELP

Java - 124 bytes

int p(int a,int b){int c=0,t=a;double p;while(t>0){if(a-(p=Math.pow(t,b))>=0){while((a-=p)>=0){c++;}a+=p;}t--;}return c+=a;}

Ungolfed

int p(int a, int b){
    int c=0,t=a;
    double p;
    while (t>0){
        if (a-(p=Math.pow(t, b))>=0){
            while((a-=p)>=0){
                c++;
            }
            a+=p;
        }
        t--;
    }
    return c+=a;
}

Result

System.out.println(p(7, 2));    // 4
System.out.println(p(4,2));     // 1
System.out.println(p(7,3));     // 7
System.out.println(p(23,3));    // 9