I think the term sought after is arguments of the verb. This term doesn't only refer to the different elements in action verb constructions, but in any clausal construction. In the Original Poster's example John, Jack and money are the three arguments of the verb GIVE. Enthusiasm is not one of the arguments of the verb. There is no special grammatical relationship between give and with enthusiasm. Rather with enthusiasm is just giving extra information in the sentence. It is, therefore, an adjunct in the clause structure.
Some verbs, like the verb HAPPEN, usually only have one argument:
A verb like BET usually has four:
- Bob(1) bet the dragon(2) fifty euros(3) that he would win the contest(4).
The arguments here are, Bob (subject), the dragon (indirect object), fifty euros (direct object) and lastly the content clause: that he would win the contest (complement). The content clause there is a complement of bet, but I don't think that the function it is carrying out has a special name - unlike for example subject.
Verbs, clauses, adjectives and so forth can all be arguments of the verb. So in, for example:
- To swim can help such conditions.
to swim is an argument, the subject, even though it is a verb.
Each argument of a verb in an utterance will have a grammatical function and a thematic role. In:
- Bob was beaten by the dragon.
Bob is the subject. It has the thematic role of patient; in other words it describes the recipient of the beating action. In:
Bob is the direct object of the verb, but still has the thematic role of patient.
In short, the number of arguments that a clause has depends on the number of slots set up to be filled by the verb. These arguments may have different grammatical functions and different thematic roles. Arguments can also be of many different types of word or phrase.