Building in Chinese shipyards will do little to fix the root cause of Russian naval shipbuilding woes - chronic under-funding (exacerbated by corruption and grift). With the modern Zvezda shipyard nearing completion, the capacity (in vessel size terms) bottleneck is all but gone, and with respect to build rate of smaller hulls, even the existing infrastructure would have been fine. For an indication of what just the traditional St. Petersburg yards are capable of *if funded decently*, you need only to look at the Pr. 22220 icebreakers. Five large (35k ton), highly specialized (nuclear turbo-electric propulsion) ships at a pace of approximately one every couple of years!
And technologically, the current generation of clean-sheet Russian warships is cutting-edge. Composite material hulls or superstructures, enclosed or integrated masts (composite or aluminium construction), AESA radars, hard-kill torpedo defences, compact hit-to-kill SAMs, AUVs/USVs... Now that domestic gas turbine supply is secured, what makes anybody think Russian yards could not build the Steregushchiy and Gorshkov class at a good pace, if funded properly?
So moving construction to a Chinese builder won't defy the principle that you only get what you pay for - corruption issues will remain and Russian wages aren't THAT high. It may improve the grift somewhat but unless the Russian MoD can ensure an adequate and regular money flow, the result will be broadly the same. You simply can't build a competitive warship on an erratic shoe-string budget, not even in China.
With that out of the way, you can also legitimately question whether Russia needs carriers in the first place - unavailability of Kuznetsov has not seemed to hamper their ambitions. A more efficient use of funding would be to double down on submarines (both nuclear and conventional), which in combination with the strategic bomber fleet should provide adequate power projection. True, it lacks the soft power (i.e. coercive visual impact) of a carrier group, but the hard power (explosive tonnage on target) is not that different to the limited carrier force Russia could hope to field. Admittedly, good satellite ISR is a prerequisite, but they are working on that, with a new spacecraft generation coming up (>2m mirror diameter EO imaging and 0.4m SAR resolution).