What are the limitations to adding wings with engines to a helicopter?
A very promising British design from the 1950s used both, a rotor at low speed and a conventional wing at high speed for lift creation. This was the Fairey Rotodyne, a prototype for a flying bus to connect remote locations with a VTOL aircraft. Its rotor was tip-jet-powered and burned a mixture of fuel and compressed air bled from two wing-mounted Napier Eland turboprops. After take-off, power was switched to the propellers and the rotor was unloaded to minimize its drag at high speed. This way, both lift-creating means could be optimized for their speed domain.

You will notice that the wing is unusually  small while the rotor is unusually large. This allowed both to operate at higher efficiency than wings or rotors which are designed to work over the full speed range. However, the combination of the two weighs more than either a rotor or a wing, which reduced the possible payload of the Rotodyne. Neither was meant to fold away because that would had increased weight even more. 



Fairey Rotodyne (picture source Ed Coates collection)

In the end, it was wavering British government support rather than any technical issues which limited the Rotodyne to a single prototype.

Today, the most promising solution seems to be a coaxial rotor with a separate source of thrust; either a propeller or jet engines. The coaxial rotor is slowed down at high flight speed in order to keep tip Mach numbers below 0.9 and the lift asymmetry of conventional helicopters is no problem for a coaxial rotor design, allowing much higher flight speeds.

Folding a wing would not help much: When the wing is not needed (which is at low speed), its friction drag is low. Folding the rotor, on the other hand, makes more sense but so far is limited to a helicopter on the ground in order to reduce its storage space on aircraft carriers. Folding in flight has only been studied conceptually.



Illustration 1B from US patent application 2015/0474290 A1 by Bell Helicopter Textron Inc.