Depending on which Space Race the Chinese here are talking about.
Race to the Space - Russia Won in 1957
Race to the Manned Space flight - Russia Won in 1961
Race to the Moon - US Won in 1969
Race to Explore Planet Mercury - US Won in 1973 with Mariner 10 Probe
Race to Explore Planet Venus - Russia Won in 1961 with Venera 1 Probe
Race to Explore Planet Mars - US Won in 1964 with Mariner 4 Probe
Race to Launch Rover on Mars - US won in 2012 with Curiosity Rover
Race to Survey Mars - US won in 2001 with Mars Global Surveryor
Race to Explore Planet Jupiter - US won in 1973 with Pioneer 10
Race to Explore Planet Saturn - US, ESA and Italian Won in 1997 with Cassini Hyugen Space Probe
Race to Explore Planet Uranus - US won In 1986 with Voyager 2 flyby
Race to Explore Planet Neptune - US won in 1989 with Voyager 2 flyby
Race to Explore Dwarf Planet Pluto - US won in 2015 with New Horizon Probe flyby
Race to Discredit 9 Planets Theory - US won in 2006 (For those who don't know, the solar system have 8 planet officially)
Race to Break Escape Velocity from this Solar System - US won in 1983 with Pinoneer 10 Probe break free of Neptune
Race to Break out of Current Solar System - US won in 1983, see above
Race to the Heliosphere - US won in 2012 with Voyager 2 cross the Heliosphere
Race to Broadcast Earth material in extra-solar system. US won with Voyager 2 Cross out of Heliosphere blasting a golden record contain Image and Sound of Earth (Shame the first thing these ET heard is Chuck Berry)
Race to Built Re-entry Space Craft - Russian Won in 1957 with Soyzu Spacecraft.
Race to the Build Space Station - US, Russia Won in 1998
There aren't much other race China can win, for example, Voyager 2 is always going to be the further Probe send from earth regardless of the speed of other probe, as it exited Heliosphere, probe would enter terminal velocity (Which mean every probe travel at the same speed) However, the Race of "Furthest" exploration is up for grab as technology evolve would make the life of probe survive longer, currently the newest deepspace probe launch by human is the New Horizon Probe launch by NASA which will expected to reach E-O-M status in 2026. By then the last objective known to Human was to observe KEO 2014 MU69 which lies outside our current Solar system.
The other race left to win by China would be Colonization of Each Planet, Rover Sent to Each Planet and Moon (except Mars, and Earth moon) and Material discovery. That's probably it.
Meanwhile Enjoy some of the world best image of outer space, courtesy NASA
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Planet Neptune
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Pluto in HD
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Charon (Moon of Pluto) by New Horizon
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Juipter by Voyager 1
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KBO 2014 MU69 position as per Hubble telescope. This will turn into HD image in 2019 when New Horizon expected to fly over them and transmit their picture back to earth, this will mark the FURTHEST POINT OF EXPLORATION of human kind as of today