reference query – How to designate a class of graphs allowing only $ k $ instances of a certain induced subgraph?

Suppose that a class of graphs $ mathcal {C} $ is defined as follows:

A graph $ G $ belongs to $ mathcal {C} $ if and only if $ G $ is in agreement,
but at most k $ $ 5 $-cycles.

I am aware that the definition of $ mathcal {C} $ is contradictory. However, I am looking for a notation like

$ G in C_4 text {-free} + k * (C_5) $

Is there such a notation in the literature?