The book I'm reading mentions that under the $ZF^{--}_F$ axioms (existence, pairing, union, extensionality, separation, replacement and foundation axioms) for every class $C$, there exists a unique class $HC$ defined by the formula:
$$\forall x (x \in HC \iff x \in C \wedge \forall y \in x, y \in HC)$$
The book calls this the class of "hereditarily C sets".
As I read it, this is the formula that describes the transitive closure of the class $C$, correct? Namely, the smallest subset of $C$, denoted $HC$, that is transitive. Is the class of "hereditarily C sets" a standard terminology, does it mean the sets of $C$ that are hereditary, and if so isn't this the same as the class of sets whose subsets are also in $C$, hence the transitive closure of $C$
Finally, is it possible to construct the transitive closure of a class using the axioms of $ZF^{--}_F$?
No: the class of hereditarily $C$ sets is the largest transitive class contained in $C$, whereas the the transitive closure of $C$ is the smallest transitive class containing $C$. A hereditarily $C$ set is a set which is in $C$, and all its elements are in $C$, and all the elements of its elements are in $C$, and so on. On the other hand, the transitive closure of $C$ consists of all sets which are in $C$, or are elements of a set in $C$, or are elements of elements of a set in $C$, and so on.
For a simple example, let $C=\{\emptyset,\{\{\emptyset\}\}\}$. Then $HC=\{\emptyset\}$, while the transitive closure of $C$ is $\{\emptyset,\{\emptyset\},\{\{\emptyset\}\}\}$. You can think of these as two different ways of "making $C$ transitive": the first one does so by throwing out elements like $\{\{\emptyset\}\}$ which violate transitivity, while the second one does so by adding in the elements needed to make $\{\{\emptyset\}\}$ not violate transitivity.
The transitive closure of a class can be defined in $ZF^{--}_F$. First, for a set $x$, we can define the transitive closure $TC(x)$ as a set by recursion on $\omega$, defining $TC_0(x)=x$ and $TC_{n+1}(x)=\bigcup TC_n(x)$ and finally $TC(x)=\bigcup_{n\in\omega} TC_n(x)$. Note that since $ZF^{--}_F$ does not include Infinity, $\omega$ is just a class and not necessarily a set, and so $TC(x)$ will be just a class (definable with $x$ as a parameter). Now given a class $C$, we can define its transitive closure as $C\cup\bigcup_{x\in C}TC(x)$.