Or just read each line into a string, then deal with that string once it is in memory.
Code:
#include<iostream>
#include<string>
#include<sstream>
#include<fstream>
using namespace std;

int main()
{
    string myInputLine;

    cout<<"How many groups?"<<endl;

    // Only use this line for reading from cin
    // Then use whatever seems best to extract information from that line
    // Two examples of how to do it follow.
    // Then all that fflush(stdin) and cin.ignore() nonsense just goes away.
    getline(cin,myInputLine);

    istringstream is(myInputLine);
    int nGroup;
    is>>nGroup;

    string delGroup;
    string delTeams;

    for(int i=0;i<nGroup;i++)
    {
        cout<<"What name for Group "<<i+1<<"?"<<endl;

        getline(cin,myInputLine);

        // 1. Avoid at all possible any use of C-style char arrays
        // 2. If you must use them, avoid lying about the size of the array
        // char GrpName[300];
        // cin.getline(GrpName,30); // should have been sizeof(GrpName)

        string FileName=myInputLine;
    }

    cout<<"Do you want to clean up all files created ?(y/n)"<<endl;
    getline(cin,myInputLine);
    char YesNo = myInputLine[0];

    if(YesNo=='y'||YesNo=='Y')
    {
        cout<<"deleting all files created"<<endl;
        cout<<"deleting Group files..."<<endl;
        system(delGroup.c_str());
        cout<<"deleting Team files and Team Extended files..."<<endl;
        system(delTeams.c_str());
    }

    return 0;
}
Mixing a whole bunch of input styles is just asking for trouble. Then you end up with all sorts of random hacks to clean up the mess.