"Almost default" copy constructor (& assignment operator) in C++

If you can isolate the specific handling in a proper RAII wrapper as Igor Tandetnik suggested: go for that.

If you still need specific processing in the copy constructor and/or assignment operator (such as register the object creation/assignment in a container or log), you can group the data members that can be default copy constructed/assigned into a separate class that you use as a base class or data member, which you handle as composite, thus:

struct x_base {
  int a,b,c,d;
  std::string name;
};
struct x : x_base {
     x(const x& other)
         : x_base(other)
     {
         descr = "copied ";
         descr += name;
         descr += " at ";
         descr += CurrentTimeAsString();
         std::cout << descr << "\n";
     }
     void operator = (const x& other)
     {
         x_base::operator =(other);
         descr = "assigned ";
         descr += name;
         descr += " at ";
         descr += CurrentTimeAsString();
         std::cout << descr << "\n";
     }
     std::string descr;
};

When you later add data members that don't need specific handling, you can simply add them to x_base.