NEW YORK/WASHINGTON (Reuters) - Republican presidential hopeful Donald Trump’s campaign raised $3.1 million from donors in May, more than doubling previous monthly hauls as he began soliciting donations to battle Democratic rival Hillary Clinton. But with spending that outpaced inflows, the New York real estate magnate’s campaign began June with just $1.29 million in cash, putting it well-behind Clinton’s $42 million war chest, according to federal disclosures filed late on Monday.  Clinton’s campaign raised $26 million in May.  The figures underscore the huge money advantage Clinton is hoping to enjoy leading into the Nov. 8 election, one that could allow her a large staff and millions of dollars of television and digital ads in battleground states. Trump, who has self-funded most of his campaign and only held his first general election fundraiser on May 25, is betting he can run a race that builds on his low-spending, insurgent primary operation. Trump’s surrogates, however, have said the cash is now “pouring in” for the general election. For months the biggest cash injections into Trump’s campaign coffers were from his personal bank accounts. He has loaned his campaign $46 million since launching last year. Trump may still have several hurdles to cross before convincing deep-pocketed donors to write the kind of checks that would make him competitive with Clinton’s campaign bank account. Trump donors, allies and other Republican operatives continue to express concerns about his campaign operation, which has been dogged by internal battles, a threadbare campaign infrastructure of about 30 paid staffers, and a barely existent fundraising apparatus. On Monday, Trump fired his campaign manager, Corey Lewandowski, who had been tasked with overseeing the campaign’s fundraising arm.  The primary Super PAC supporting Trump, Great America PAC, reported raising $1.4 million in May - accounting for most of the $2.5 million the group has raised this year. The PAC had $500,971 cash remaining at the beginning of June. Clinton’s cash advantage has been fueled in part by the Super PAC supporting her. Priorities USA raised $12 million in the last month, ending May with a $52 million in cash. Three unions, AFT Solidarity, Liuna Building America and International Union of Operating Engineers, each gave $1 million. The group has largely been tasked with attacking Trump. And so far, they have spent more than $5.7 million this year on television ads alone attacking the Republican. (In fifth paragraph, this story corrects the date of first Trump general election fundraiser to May 25) 