- "Cough" is a nonfinite verb form. That means that it's not the predicate of a clause. English has three of these -- infinitives ("to" form), gerunds (-ing form), and participles (-ed form, for regular verbs anyway). These all have a verb-like meanings of doing or being, but they act as other parts of speech in the sentence.
"Cough" is called a bare infinitive because it's lost the "to". Some verbs require a "to" infinitive:
I want to go. (Not, I want go.)
and some require a bare infinitive:
I must go. (Not, I must to go.)
Some verbs allow either an infinitive or a gerund
I like to eat (Or alternatively, I like eating.)
You just have to learn which verbs take which.
In this example, an equivalent phrasing would be "beggar coughing with a sickness." Either way, the "cough" verbal form modifies beggar.
- "You knew" is a relative clause that has elided the relative conjunction. It modifies sickness, telling us which sickness. It means
a sickness [that] you knew.
- "would kill him" is another noun clause with an elided relative conjunction "that" serving as the subject of "would kill". It's antecedent is "sickness." The whole clause is what you knew so the whole clause is the direct object of "knew."