When Newt Gingrich’s campaign disclosed in October it planned to pay the candidate $70,000 for travel expenses in the third quarter, the transfer was an anomaly among presidential campaigns. But weeks ago, the former House speaker revised his third-quarter bill: He actually expected to personally receive $115,000 of the funds donated to his campaign to reimburse himself for expenses during that period. The campaign would not explain how the candidate forgot about $45,000 in receipts, what they were for or why the campaign wasn’t simply paying the invoices directly, as other campaigns do. Far beyond that payment, the uses of...