The Yak-44 is not going to be pushed back into development anytime soon, no matter what the commi's say.
First off, the Kuznetsov does not have catepults, which are needed for the E-2 & Yak-44. Sure, it has a ski jump, but theres simply not enough actual runway for the Yak-44 to speed up and utalize the ski jump on the Kuznetsov. The Su-27K Flanker's can do the ski jump easily, but remember, they have two jet engines, not two props. This is one of the obvious and main reasons for the United Kingdom not having some varient of the E-2 for their aircraft carriers.
They're is simply not enough room aboard the Kuznetsov to actually house the Yak-44. They have trouble holding the 12 Su-27K's that it was designed to carry, imagine trying to also fit in atleast two Yak-44's. Taking away hanger space is the price one pays for sticking Vertical Launch Tubes in the flight deck. Number wise, the hanger bay is only about half the size of the flight deck on the Kuznetsov.
The Yak-44 was intended to be stationed aboard the Ul'yanovsk Class carrier. This carrier - originally designed in 1973 - was to try and compete with Forrestal - Kitty Hawk - Enterprise - Kennedy - Nimitz class carriers by carring 70+ aircraft ranging from the Su-27K, Su-25, Yak-44, and 15 - 20 helicopters. It was inteded to have a ski-jump on the bow similiar to the Kuznetsov but also was intended to have two catepults on the angled flight deck which were obviously to launch the Yak-44 and aircraft that had loadouts to heavy for the ski jump.
You can see this in one of the builders models for the Ul'yanovsk Class.

- Chris