* SSEC +0.7%, CSI300 +0.8%, HSI +0.5%
* China c.bank cuts medium-term loan rate for 1st time since 2016
* China presses Trump for more tariff roll-backs in 'phase one' trade deal
SHANGHAI, Nov 5(Reuters) - China and Hong Kong stocks firmed on Tuesday, as investors cheered Beijing's latest policy easing to boost the economy after a private survey showed sluggish growth in the country's services sector.
** The CSI300 index rose 0.8% to 4,009.82 points at the end of the morning session, while the Shanghai Composite Index gained 0.7% to 2,995.66 points.
** The Hang Seng index firmed 0.5% to 27,694.49 points, while the Hong Kong China Enterprises Index climbed 0.7% to 10,889.35 points.
** The Shanghai SE 50 index, dubbed China's "Nifty 50" index and tracking the 50 most representative heavyweights on the Shanghai Stock Exchange, climbed as much as 1.0% to its highest since February 2018.
** China's central bank cut the interest rate on its medium-term lending facility on Tuesday for the first time since early 2016, as policymakers work to prop up a slowing economy hit by weaker demand at home and abroad.
** "Beijing will likely introduce more policy easing measures in coming quarters to stabilise market sentiment and bolster growth, but the space is much more limited than in previous easing cycles, especially with surging CPI inflation," analysts at Nomura said in note.
** The cut came after a private survey showed China's services sector activity expanded at its slowest pace in eight months in October as new orders slowed and business confidence hit a 15-month low.
** Beijing has been counting on the services sector, which accounts for more than half of China's economy, to partly offset sluggish domestic and global demand for manufactured products as a prolonged trade war with the United States drags on.
** Developments in Sino-U.S. trade talks remained in focus.
** China is pushing U.S. President Donald Trump to remove more tariffs imposed in September as part of a "phase one" U.S.-China trade deal, people familiar with the negotiations said on Monday.
** Global trade barriers must be removed, and countries should uphold basic multilateral trade principles while standing firm against protectionism and unilateralism, Chinese President Xi Jinping said in a speech on Tuesday.
** Around the region, MSCI's Asia ex-Japan stock index was firmer by 0.35%, while Japan's Nikkei index was up 1.98%.
** The yuan was quoted at 7.0224 per U.S. dollar, 0.12% firmer than the previous close of 7.0305.
** So far this year, the Shanghai stock index is up 19.31%, while China's H-share index is up 6.8%. Shanghai stocks have risen 1.59% this month.
** As of 04:03 GMT, China's A-shares were trading at a premium of 28.76% over the Hong Kong-listed H-shares. (Reporting by Luoyan Liu and John Ruwitch; Editing by Aditya Soni)