How to remove element not at top from priority_queue?

The standard priority_queue<T> can be customized through inheritance. It has protected members c and comp that can be referenced in a descendant class.

template<typename T>
class custom_priority_queue : public std::priority_queue<T, std::vector<T>>
{
  public:

      bool remove(const T& value) {
          auto it = std::find(this->c.begin(), this->c.end(), value);
       
          if (it == this->c.end()) {
              return false;
          }
          if (it == this->c.begin()) {
              // deque the top element
              this->pop();
          }    
          else {
              // remove element and re-heap
              this->c.erase(it);
              std::make_heap(this->c.begin(), this->c.end(), this->comp);
         }
         return true;
     }
};

void main()
{
   custom_priority_queue<int> queue;

   queue.push(10);
   queue.push(2);
   queue.push(4);
   queue.push(6);
   queue.push(3);

   queue.remove(6);

   while (!queue.empty())
   {
      std::cout << queue.top();
      queue.pop();

      if (!queue.empty())
      {
        std::cout << ", ";
      }
   }

 }

Output:

10, 4, 3, 2