In the mammalian nervous system, various synaptic plasticity rules act, either individually or synergistically, over wide-ranging timescales to enable learning and memory formation. Hence, in neuromorphic computing platforms, there is a significant need for artificial synapses that can faithfully express such multi-timescale plasticity mechanisms. Although some plasticity rules have been emulated with elaborate complementary metal oxide semiconductor and memristive circuitry, device-level hardware realizations of long-term and short-term plasticity with tunable dynamics are lacking. Here we introduce a phase-change memtransistive synapse that leverages both the non-volatility of the phase configurations and the volatility of field-effect modulation for implementing tunable plasticities. We show that these mixed-plasticity synapses can enable plasticity rules such as short-term spike-timing-dependent plasticity that helps with the modelling of dynamic environments. Further, we demonstrate the efficacy of the memtransistive synapses in realizing accelerators for Hopfield neural networks for solving combinatorial optimization problems.