This paper discusses the effects of radiation damage to SiPMs on the performances of plastic scintillator counters with series-connected SiPM readout, focusing on timing measurements. The performances of a counter composed of a $120 times 40 times5~mathrm{mm}^3$ scintillator tile read out by two sets of six SiPMs from AdvanSiD connected in series attached on the short sides are presented, for different combinations of SiPMs at various levels of irradiation. Firstly, six SiPMs were equally irradiated with electrons from $^{90}$Sr sources up to a fluence of $Phi_mathrm{e^-}approx 3 times 10^{12}~mathrm{cm}^{-2}$. The timing resolution of the counter gradually deteriorated by the increase in dark current. The dark current and the deterioration were reduced when the counter was cooled from 30$^circ$C to 10$^circ$C. Secondly, 33 SiPMs were irradiated with reactor neutrons. The characteristics of counters read out by series-connected SiPMs with non-uniform damage levels, were investigated. The signal pulse height, the time response, and the timing resolution depend on the hit position in the counter, when SiPMs irradiation is not uniform.
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