MEXICO CITY, June 25 (Reuters) - Mexican state energy company Pemex upped the volume of crude oil it exported in May, following a substantial dip a month earlier, and returned to levels closer to its monthly average, official data showed.

Pemex exported 910,801 barrels per day (bpd) in May, up 34% from April, after it reversed export cuts it had originally planned to be able to feed its six domestic refineries and the Olmeca refinery in Dos Bocas.

Its sudden reversal of export cuts for May came after Pemex had made substantial cuts for April that did go ahead.

Data published by the company late on Monday also showed the combined processing of its six local refineries in operation fell in the fifth month of the year to 842,043 bpd, its lowest level this year.

Even after billions of dollars of investment under current President Andres Manuel Lopez Obrador, heavily-indebted Pemex is struggling to maintain the volume of crude oil it refines.

CEO Octavio Romero said on Friday that the new Olmeca refinery would work at full capacity next month.

However, sources told Reuters that engineers were still working on key components of the 340,000 bpd refinery, and that it was far from ready despite multiple promises.

With less being processed at home, Pemex increased its imports of liquefied natural gas, gasoline, diesel and others in May to a total of 670,000 bpd, 11% more than in April.

Pemex produced 833,280 bpd of refined petroleum products, which includes gasoline and diesel but also a lot of fuel oil, 11% less than in April and the lowest this year.

Crude oil and condensate production averaged 1.79 million bpd for the month, lower than the average for the year so far, while crude oil production increased slightly to 1.51 million bpd. (Reporting by Ana Isabel Martinez Editing by Stefanie Eschenbacher and Marguerita Choy)