Heat can be considered as an inexact differential. Hence, we could conclude that: there exists a path $\gamma$,such that: $$ \oint\delta Q \ne 0 $$ Hence, $\int\delta Q\ne\Delta Q$.
i.e. $\Delta Q$ makes little to no sense.
Yet, I have seen equations where $\Delta Q$ is used. E.g. Latent Heat.
Moreover, I have seen situations where people treat this $\delta Q$ is treated as $d Q$ while writing some equations.
Is this justified? If so, how? Under what conditions?
The notation used for heat in thermodynamic processes is one of the poorest in Physics. A source of confusion is that the total exchanges of heat (Q) and work (W) always sum to the internal energy difference between the final and the initial state. Therefore, one should write: $$ \Delta U = Q + W \tag{1} $$
A notation like $$ \Delta U = \Delta Q + \Delta W \tag{2} $$ It is incorrect but could be accepted, provided it is clear that the $\Delta$s on the right-hand side of the equation do not represent differences but increments of heat and work exchanged with the system. That is the meaning that should be assigned to the notation for latent heat in Clapeyron's equation. Altogether, I find the form ($1$) clearer.
For small changes in the state, the difference between $\Delta U$ and the differential $dU$ can be neglected. In many textbooks of the beginning of the twentieth century (Planck's textbook for sure), the differential form of equation $(1)$ was written as $$ d U = q + w \tag{3} $$ that I find coherent with the fact that the symbols on the right-hand side of the equation, in general, are just two real numbers and do not represent variation or differential of anything. For an irreversible process, they do not even depend on state variables. Of course, in the case of reversible processes, it is possible to give an expression of $q$ and $w$ as $TdS$ and $-pdV$, respectively, but this is a special case.
Unfortunately, such a terse notation has been forgotten, and all kinds of $d-$like symbols have been introduced to represent $q$ and $w$, like $\delta q$, or a barred $d$. Even worse, the term inexact differentials is often used for such quantities, introducing additional confusion between the genuine mathematical concept of non-exact differential forms and some quantities that are not a function of the state variables, in general.
To summarize, the only case where it is justified to use a notation like $dq$ is in the case of reversible processes, where it is a shortening for the differential $TdS$.
Since all infinitesimally transported heat $\delta Q$ is also some amount of entropy $dS$ transported at some temperature $T$, that is, $\delta Q = TdS$, you can always sum (integrate) their product over some path without any problem: $Q[\gamma]=\int_{\gamma} TdS$ where the integral is now the total heat $Q[\gamma]$ transported over the path. Here $Q[\gamma]$ depends on the path and not just on the endpoints and any "point" of the parametrized path $\gamma$. Because of their implied continuity $T,S$ is well defined by the interface through which the transport occurs.
