Is there a studied model of Petri nets where each transition only has one normal in-edge (which removes a marker from a place), but can also have "checking" edges, which only check whether there is a marker in a place?
Alternatively, each transition has at least one out-edge to each of its input places, except for one.
Such a net would model a random-permitting context grammar, but a Google Scholar search for "random permitting context" and "petri net" doesn't yield any likely candidates.
EDIT: it seems my question is not very clear, but I'm unsure exactly of how to improve it.
I'm looking for a restricted class of Petri nets: these nets can indeed be simulated by ordinary Petri nets, so I'm definitely not looking for a generalisation of Petri nets.
Unless I'm more-than-usually confused, there are Petri nets which cannot be simulated by these nets.
Perhaps my motivation and what I mean by my definition can be clarified by a further restriction: we keep the restriction of each transition having only one normal in-edge, but further disallow checking edges. In this case, we get a model (the Parikh vectors of) the context-free languages: each transition represents a production and places represent symbols, with a production deleting one symbol and adding any number of others.
Marek and Češka have shown that Petri nets can simulate any random-context language; the random-permitting context languages are a strictly smaller class (even when we ignore the order of symbols). For example $\{a^{2^n} : n \in \mathbb{N}\}$ is a random-context language but is not random-permitting context language (these languages satisfy a pumping lemma).
The Petri nets here correspond to these languages: the production $A \rightarrow XYZ$ with permitting set $\{C,D\}$ corresponds to a transition with an in-edge from $A$, out-edges to $X$, $Y$ and $Z$, and checking edges from $C$ and $D$.