For some reason, Brandon Blatcher
's comment to the question got downvoted into oblivion, but it should not have, because it is the correct answer.
NASA is always interested in lunar missions. NASA is always interested in Martian missions. NASA is always interested in asteroid missions. NASA is always interested in Jovian missions. NASA exists to explore.
But NASA doesn't decide where it goes. The President and Congress decide.
So the more accurate question is - If China lands on the moon, is that likely to increase Washington's interest in lunar missions?
And the answer to that is, maybe.
There are 536 different opinions on that matter (1 President, 100 senators, and 435 congressional representatives). For the most part, none of those people are very interested in space. The only ones that treat it as an issue worthy of their attention are the ones that have space jobs in their district/state.
In the 1960's there was not universal support to go to the moon. But there was near universal fear of appearing to side with "the commies". The US and the Soviets were in a battle for supremacy across many fronts to prove which form of governance was superior. It was important to the US that they be seen as clearly better than the Soviet Union. The Russians were kicking America's rear-end at space. America desperately needed a victory so they put their eggs in the basket of going to the moon.
There isn't that same level of competition, today, between the US and China. They aren't mortal enemies. They depend on each other. China practically funds the US and the US purchases China's goods.
While I think that having China as a strong competitor in space would elevate the level of discussion about space, and therefore possibly make it a topic of interest to politicians, I do not believe it would be enough to kickoff a space race, again.