Going from gas to plasma is a phase transition when the thermal motion of the atoms is so great that when atoms collide they shake off electrons (ionization). More precisely, a phase transition occurs when thermal energy is higher than the ionization energy.
However there is no plateau in the heating curve, because there is no latent heat of ionization. Not every change of phase necessarily involves a latent heat! A phase change that does involve a latent heat, such as melting and boiling, is called a first-order phase transition. A phase change that doesn't involve a latent heat is called a second-order phase transition. Some second-order phase transitions are: normal-conductor to superconductor, nonmagnetic to magnetic, and liquid to glass.
During a first-order phase transition, a property of the substance makes a sudden, abrupt change. What that property is depends on the specific transition. For example, when a liquid turns to gas, its density suddenly changes from high to low.
During a second-order phase transition, a property of the substance goes from zero and steadily increases. For gas-plasma transition, one such property is electrical conductivity. A gas has zero conductivity, but when it turns into a plasma, its conductivity steadily grows but doesn't jump.
These are rather subtle issues for the high-school level, which is why I think the textbooks avoid it altogether.

I think that you are asking about the specific heat of a gas as it is heated to a temperature where it becomes a plasma due to dissociation of some electrons.
Essentially you have gas molecules moving faster and faster (as well as vibrating and rotating harder) until there is some realistic probability that somewhere in the material, you get a high enough energy collision to scatter an electron away from a molecule. If there are sufficient collisions, then the number of scattered electrons (before they recombine) is constant. At this point, if you look at the system,you now have more particles (both electrons and Ions) each of which can carry energy so to further increase the temperature (or energy per particle) there are more such particles so it takes more energy. Actually, in practice with a molecular gas, a phase change is noted when the gas molecules dissociate to make a partially atomic gas -- i.e. O₂--> O + O which causes a large plateau as well. -- Upshot, the notion of a phase change in relation to specific heat is characterized by some mechanism to redistribute thermal energy -- i.e.
from vibration modes to liquid phase (still adjacent by randomly oriented) to gas phase (simple kinetic model + local energy storage in rotation and vibration) to gas dissociation to ionization and several more at higher temperatures such as second ionization, third ionization, etc.
Every such change has a plateau in the specific heat. The essential issue is the distribution of energy into whatever states are available and if there are more particles -- there are more states. Also if there are different kinds of processes that can occur (i.e. ionization) to store energy, the heat will plateau.


Such a process can work both ways -- i.e. a dissociation can be energetically favorable and increase the number of states so much that the temperature actually decreases...
Like dissolving potassium iodide in warm water -- leaves you with cold brine.

I believe you are correct in that the transition from a gas to a plasma (ionized gas) is not an abrupt process like with other phase changes, but rather occurs gradually with increasing temperature. Therefore, there would not be a plateau in the heating curve