Well, comparing with the CFM56 is a bit of a misnomer, the engine core dates back to the 1970s from the GE B-1 bomber engine core.
There are much more modern engines available right now. Even the Russians have better engines than the CFM56 by this time.
I doubt the Chinese engine shares much with the CFM56. They might have started with that as a base but I doubt it hasn't changed significantly. I think most significant improvement they had was with the turbine blades. They finally got relatively modern technology for those. It seems they got the technology from Germany? And now they are supplying the Germans with turbine blades.
Europe is ever more pathetic.
I think China has the potential to eclipse Russia with the use of CMC (Ceramic Metal Composites) in the hot sections of the engine in the near future. This means the engines would be able to run at much higher core temperatures and thus have increased efficiency. But the Russians have always had a lot of tricks with metallurgy so who knows they might be able to cope even with that.
Remember the MiG-25 was made to intercept Valkyrie bombers and the SR-71. The SR-71 used all titanium construction. The Valkyrie used a honeycomb metal meta-material that never quite worked properly without cracks and deformations (this was one of the reasons the project was eventually canned), the Russians? They just used steel with nickel coating. Sure it's heavy, but you put two HUGE engines on it and make it look like a brick with just the minimum wings you can put on it, guess what it does the job. Even Elon Musk, who started with fancy carbon composites for his Starship project, remember this is a spacecraft which is to fly at high Mach speeds, switched to really thin and hard steel alloys.
I think the next Chinese military engines will have CMCs and perhaps cooling channels, if they don't have already, and I suspect they will be able to operate at high Mach regimes with a variable intake of some form.
I am currently interested in what the heck the Russians are doing with the PAK-DP. That is supposed to be the aircraft the SR-71 successor never was. Given their budget restrictions other countries might have just manufactured more Su-57s. But this is Russia. I suspect they will make a high-speed long distance interceptor and perhaps even bomber aircraft. This is a big ticket item I think China also needs to have and make the technology for this. If you make it large enough, think Tu-22M sized, it could be a superb AA/AD weapon against all sorts of air and naval threats. This will require all new engine technologies. The Russians are also investigating pulse-detonation engines. But I doubt we'll see one of those anytime soon.