Without Mao, the CCP loses the person who kept it independent and successful. Most of the other CCP leaders believed in revolution based on the proletariat, as opposed to farmer. The CCP always kept getting destroyed fighting Chiang in that way, but its what orthodox Communism told them to do. Furthermore, Mao was also the one most insistent on focusing on defeating Chiang as opposed to the Japanese. This was important in retaining the CCP's strength for the postwar struggle as well as killing any chance of reconciliation with the Nationalists after the war.
In 1939, the Nationalist Army was still fairly capable. It had not been degraded in the fiasco of the 39/40 Winter Offensive nor suffered years of isolation from the Burma Road being cut off. It would win several important victories before the end of the year, and still retained a lot of public support.
With Mao dead, I think the CCP will cooperate much more with the Nationalists in the United Front. This is what Mao's rivals all wanted to do (as well as some of Mao's allies), and with him gone, they are likely to succeed. CCP armed forces will follow KMT orders and be more willing to fight the Japanese. This will both help the Chinese war effort, but degrade the CCP's forces at the same time. None of this will materially affect the outcome of the Sino-Japanese War, but it will pave the war for postwar reconciliation.
After the Japanese surrender, the US will press the KMT to include the CCP into the next government. Chiang didn't like that idea, but agreed IOTL. ITL, he will have a much better experience with the CCP during the war and probably forged every closer ties with Zhou Enlai and other top Communists. While this is very debatable, I think the KMT and CCP will be able to make it work without Mao. If the CCP agrees to incorporate its forces with the Nationalist army, Chiang will agree to include the CCP under American pressure. At which point, the more reasonable CCP reforms will be enacted and Chiang will begin other reforms aimed at centralizing his authority and reducing the power of the warlords.
There will be essentially be 4 factions in China. The pro-Chiang cadre. The warlord/anti-Chiang faction within the KMT. Liberal reformers who vary between being pro and anti Chiang. And finally the Communists who probably don't like Chiang, but like some of his enemies even less. Eventually the warlord faction will collapse, with some going to Chiang and others joining the liberal reformers. CCP influence will likely be at a high point right after the war, but under the burden of being orthodox Communists, they won't make much headway.
It's possible that the Soviet Union retains a lot of influence in Manchuria, and that Manchuria even becomes its own de facto (or even dejure) state without Mao constantly badgering Stalin to give it up. In any case, actual KMT control will remain weak with Stalin keeping lots of extraterritorial powers in Manchuria, and a subservient local Communist party. It is possible Chiang gives up any chance to control Manchuria because of the difficulty. This will hurt the popularity of the CCP soon after, keeping them a controllable threat. There may even be a split between those who retain nationalist credentials and stay with Chiang, and those who eventually leave to become the leaders of a Soviet Manchuria who will retain the same level of real independence like Eastern Europe.
The Manchurian question has a lot of things to consider. Scenarios seeing Manchuria staying as part of China, splitting off as a Soviet puppet state, or leading to an eventual war in the 1950s if Chiang thinks China is ready to conquer and control it are all possibilities.