Are there Petri net models with "checking" rather than "taking" transitions? 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 $D$.
What you are looking for with "checking edges" are known as Petri nets with read arcs.  In your question you also mention nets with a single read edge, those are called communication-free.  It would be difficult (for me) to point to the original author(s) for these notions, but they are fairly standard.  Relationships between communication-free Petri nets and Parikh images of context-free languages are investigated for instance by Javier Esparza, Petri Nets, Commutative Context-Free Grammars, and Basic Parallel Processes, FCT 1995, LNCS 965, pp. 221--232, and Fundamenta Informatica 31(1):13--25, 1997.