In order to achieve the ultimate goal of reducing coincidence time resolution (CTR) to 10 ps, thus enabling reconstruction-less positron emission tomography, a Cherenkov-radiator-integrated microchannel plate photomultiplier tube (CRI) reaching CTR of sub-50 ps full width at half maximum (FWHM) has been developed. However, a histogram of time differences between a pair of the CRIs shows undesirable side peaks, which are caused by gamma rays directly interacting with the micro channel plates (MCPs). Such direct interaction events are detrimental to the timing performance of the CRI. In this paper, we demonstrate an analytical method of deconvolving MCP direct interaction events from the timing histogram. Considering the information of the main and the two side peaks, the timing uncertainty caused by the MCP direct interaction events is deconvolved and the CTR of the CRI is analytically investigated. Consequently, the CTR is improved from 41.7 to 40.5 ps FWHM by the deconvolution. It means that a mixture of the Cherenkov radiator events and the MCP direct interaction events contribute to the CTR by a factor of 10 ps. The timing performance of the MCP direct interaction events are also evaluated. The CTR between the two MCPs is found to be 66.2 ps FWHM. This indicates that a photocathode-free radiation detector with high timing performance is possible. Elimination of the photocathode from the detector would make detector construction easier and more robust.