The compound gerundio
The gerundio composto is made of avendo or essendo plus the participle. It shows that the action was complete before the main clause. Usually it also implies a reason.
Blueprint
At the same time or before
Simple: at the same time
Leggendo il giornale, ho scoperto lo sciopero.
Reading the paper, I found out about the strike.
Compound: before
Avendo letto il giornale, sapevo già dello sciopero.
Having read the paper, I already knew about the strike.
The auxiliary is the same as in the passato prossimo. With essendo the participle agrees with the subject. Pronouns attach to the auxiliary.
Essendo arrivata tardi, Giulia ha perso l'inizio.
Having arrived late, Giulia missed the beginning.
Avendolo già visto, sono rimasto a casa.
Having already seen it, I stayed at home.
Pur avendo studiato molto, non ha superato l'esame.
Despite having studied hard, he didn't pass the exam.
The compound gerundio sounds written. In conversation you usually say Siccome avevo già letto il giornale, … or Dato che era arrivata tardi, ….
The gerundio with its own subject
As a rule the gerundio shares the subject of the main clause. In formal texts it can have its own subject. That subject then comes after the gerundio.
Essendo il treno in ritardo, abbiamo preso un caffè.
The train being late, we had a coffee.
Mancando il direttore, la riunione è stata rinviata.
With the director absent, the meeting was postponed.
Stando così le cose, non possiamo fare altro.
Things being as they are, there's nothing else we can do.
In the set phrase tempo permettendo (weather permitting) the subject comes first by exception.
Condition and gradual change
The gerundio as a condition: if
Before a main clause in the present, future or conditional, the gerundio often means if. The compound gerundio with a past conditional stands for a missed condition.
Volendo, si può prenotare anche online.
If you like, you can also book online.
Prendendo il treno delle otto, arrivi in tempo.
If you take the eight o'clock train, you'll be on time.
Avendolo saputo prima, sarei venuto anch'io.
Had I known earlier, I'd have come too.
andare + gerundio: little by little
andare with a gerundio describes a development that moves forward step by step. It sounds elevated and is typical of reports.
La situazione va migliorando.
The situation is gradually improving.
I prezzi vanno aumentando di mese in mese.
Prices are going up month by month.
English having done and absolute phrases like the train being late map closely onto these forms. The difference is register: Italian uses them more freely in writing, while in English the absolute phrase often sounds old-fashioned.