We use the deuteration of $c$-$\mathrm{C_{3}H_{2}}$ to probe the physical parameters of starless and protostellar cores, related to their evolutionary states, and compare it to the $\mathrm{N_{2}H^{+}}$-deuteration in order to study possible differences between the deuteration of C- and N-bearing species. We observed the main species $c$-$\mathrm{C_{3}H_{2}}$, the singly and doubly deuterated species $c$-$\mathrm{C_{3}HD}$ and $c$-$\mathrm{C_{3}D_{2}}$, as well as the isotopologue $c$-$\mathrm{{H^{13}CC_{2}H}}$ toward 10 starless cores and 5 protostars in the Taurus and Perseus Complexes. We examined the correlation between the $N$($c$-$\mathrm{C_{3}HD}$)/$N$($c$-$\mathrm{C_{3}H_{2}}$) ratio and the dust temperature along with the $\mathrm{H_2}$ column density and the CO depletion factor. The resulting $N$($c$-$\mathrm{C_{3}HD}$)/$N$($c$-$\mathrm{C_{3}H_{2}}$) ratio is within the error bars consistent with $10\%$ in all starless cores with detected $c$-$\mathrm{C_{3}HD}$. This also accounts for the protostars except for the source HH211, where we measure a high deuteration level of $23\%$. The deuteration of $\mathrm{N_{2}H^{+}}$ follows the same trend but is considerably higher in the dynamically evolved core L1544. Toward the protostellar cores the coolest objects show the largest deuterium fraction in $c$-$\mathrm{C_{3}H_{2}}$. We show that the deuteration of $c$-$\mathrm{C_{3}H_{2}}$ can trace the early phases of star formation and is comparable to that of $\mathrm{N_{2}H^{+}}$. However, the largest $c$-$\mathrm{C_{3}H_{2}}$ deuteration level is found toward protostellar cores, suggesting that while $c$-$\mathrm{C_{3}H_{2}}$ is mainly frozen onto dust grains in the central regions of starless cores, active deuteration is taking place on ice.
