InGaAs/InP devices suitable as Single-Photon Avalanche Diodes (SPADs) for photon counting and photon timing applications in the near-infrared provide good detection efficiency and low time jitter, together with fairly low darkcount rate at moderately low temperatures. However, their performance is still severely limited by the afterpulsing effect, caused by carriers trapped into deep levels during the avalanche current flow and later released. We present preliminary experimental characterization of recently-developed InGaAs/InP detectors that can promisingly be operated slightly cooled. We investigate the primary dark-count rate, taking into account both thermal generation in the InGaAs absorption layer and trap-assisted tunnelling in the InP multiplication layer. We report on improvements obtainable by selecting the proper operating conditions and electronic circuit solutions. The fundamental role played by the front-end circuits in minimizing the effects of afterpulsing is assessed and demonstrated. We report the performance of a 25 μm-diameter InGaAs/InP SPAD at 1550 nm wavelength, with dark count rate of 400 cps (count per seconds) at 175 K and just 2000 cps at 225 K, with afterpulsing showing off only below T_OFF=10μs. The photon timing resolution is 100 ps (FWHM, Full Width at Half Maximum) at 7 V of excess bias.