supports pressing ahead???

press ahead with是連在一起的
請見：http://www.google.com.tw/dictionary?q=press+ahead%2Fon+(with+something)&hl=zh-TW&sl=en&tl=zh-TW&oi=dict_re
意思是 繼續.... 繼續進行......

She said China has taken its own measures to fight climate change and supports pressing ahead with international cooperation.
所以可以譯為：她表示中國已採取自己的措施來對抗氣候變遷，同時支持繼續推動國際合作。

若要拆成兩句，可以拆成China has taken its own measures to fight climate change

China supports pressing ahead with international cooperation.
support加s是因為主詞是中國，單數，要加s
該句已經有support為動詞了，所以press ahead with這個動詞片語，press要加ing變動名詞。

希望這樣對你有所幫助嚕!加油!

依據文義，我認為你對的應該沒錯，support應是與fight為對等句，所以不應加s。(可以拆成兩句沒錯但較為重複冗長)

pressing ahead with international cooperation 應為分詞構句，用來修飾support，意思就是與國際合作共同對抗 (氣候變化)。