Yes, as they have since the war, just as Taiwan claims China. Your point?

Peace with Taiwan has been maintained for nearly a hundred years, with a mutual understanding that nobody would try to force the issue too hard (look up "strategic ambiguity"). In recent years, the US has been recklessly deviating from that understanding and now people treat the status quo as "Chinese aggression," because of propaganda.

There is no "claim" to be made and there is no "reclaim" to be had against such a claim.

Taiwan is and always has been recognized as part of the country of China. That's why the losing army in the civil war went there - because it was part of the country they were a party of.

China has stated for 70 years that the island province of Taiwan will be integrated into the rest of the governance of the country. For 50 years it has explicitly stated it will be integrated peacefully, because the CPC recognizes that doing it forcefully would actually be contradictory and create a constant guerilla warfare situation as well as invite the world's militaries to intervene. The CPC has no intention of forcing Taiwan to integrate except if Taiwan works with foreign governments to establish a substantial and real threat to the security of the mainland.

If China waits long enough, the Western economies will collapse and Taiwan will very quickly and easily realize that the West just can't support them anymore and when they look to see who they depend on for nearly everything, and who their relatives are and who their dominant trading partner and who can protect them militarily, it's going to be an easy process of integrating the provincial government of Taiwan into the government of the mainland - especially since the CPC is committed to One Country Two System meaning the provincial government of Taiwan can continue operating with the same structure and same politicians and same processes as it has now.

China seeks to reclaim Taiwan as part of China for the same reason Taiwan seeks to reclaim the mainland as part of China.