I want to update my submenu item as checked when the password entered is correct in MFC.

CheckMenuItem is a CMenu class function. So you need a pointer your menu.

If the item is inside a submenu of your main menu you can use the MFC command handling in your CDocument derived class instead:

ON_UPDATE_COMMAND_UI(ID_OF_THE_MENU_ITEM, OnUpdateAllowEditing)

void CMyDoc::OnUpdateAllowEditing(CCmdUI *pCmdUI)
{
    //pCmdUI->Enable(m_nAllowEdtining);
    pCmdUI->SetCheck(m_nAllowEdtining);
}

This is the preferred method because you don't have to retrieve a pointer to the main menu which may be of different types (CMenu or CMFCMenuBar).

If the item is part of a dynamically created menu (e.g. a popup or context menu) you have probably stored a pointer to that menu somewhere. Then call GetSubMenu(0) to get a pointer to the sub menu which can then be used to call CheckMenuItem().