The supply system in HAN is the so called "simple" system. Armies draw supply from the regions they are in, and from adjacent regions, if controlled. If said regions have depots or other structures that can provide extra supply, the better.
Fleets do not "give" supplies to neighbouring land forces.
Supply carts (impedimenta) can draw supplies from supply sources and transport them to armies in far away regions.
So, what you must do if you want to carry a sucessful siege of Pharos, seeing that Illyria regions give very little intrinsic supply:
- load your army (with fully loaded impedimenta)in the fleet and unload in Pharos.
- start the siege.
- sail your fleet to Italy and load some full-supplied impedimenta in them (that you started to build on first turn, of course

).
- sail back to Pharos coast, unload your fresh supply carts and rotate your almost depleted carts to the fleet.
- sail back again to Italy, to replenish your supply carts.
etc.
Not easy, but realistic, I would say.
(and much more difficult for the AI to conduct such supply-train manouevres)
Hope it helps
(PS: regarding cities assaults: in general, you can only assault a city with a "wall" level of one, or, in case of higher fortification levels, when a suficient level of breaches have been made in the walls( in that case the siege icon turns red and the tooltip will say that you can assault).
Some rare and special leaders can assault any city (they have the "assaulter" ability, like Julius Caesar); also, in siege warfare in AJE games, romans have a distinct advantage as their legions can automatically build siege engines (although republic-time legions like in HAN with a much lower chance that empire ones). Finally in some scenarios you can have some RGD "cards" of siege works, that you can use to breach walls. Sucess is not guaranteed and the number of said siege cards is limited - and will not regenerate. For example, Carthage starts with a limited number of said cards, in the SPW scenarios)