I’m not sure what source you’re using but I think this is several incidents from the Third Crusade mixed together.

First, Richard sailed from Sicily to the east but his fleet ran into a storm and he had to land on Cyprus, which was a province of the Byzantine Empire at the time, although it was actually ruled by a rebellious lord, Isaac Komnenos. Isaac took Richard’s treasury and took some of Richard’s family hostage, and in the end, Richard ended up conquering the whole island. It was then ruled by a crusader dynasty for the next several hundred years.

Richard continued on to Acre where he joined up with Philip II of France and the other crusaders. They captured Acre, but Richard managed to offend everyone in various ways and Philip went back to France.

After taking Acre, Richard and Saladin tried to negotiate the release of each other’s prisoners of war, but they weren’t able to come to a satisfactory agreement so Richard executed about 3000 prisoners. Richard blamed Saladin for stalling, and the Christian side generally felt this was revenge for Christian soldiers being killed during the siege. Saladin then pretty much had no choice but to execute all his prisoners as well, so no one got any prisoners back.

This is where your question comes in. The crusaders could be resupplied by sea now that they held Acre, and there wasn’t much Saladin could do about that. But Saladin wasn’t just going to go away since he had access to all the resources of the surrounding area. On the other hand, Saladin also knew that his men would want to go home eventually and he couldn’t keep an army in the field forever. So both sides apparently thought that they would have to defeat the other side in a full open battle if they wanted to accomplish anything.

So Richard marched south from Acre towards Jaffa, where he could (hypothetically) turn east and continue to Jerusalem, or (more likely) meet Saladin somewhere in between. The army marched along the coast and whenever they came to a port, supply ships docked to feed them. Saladin’s troops followed them and occasionally harassed them; at one point Richard led a cavalry charge against a small group of Saladin’s cavalry. But Saladin didn’t risk a pitched battle with his full army until just outside the town of Arsuf, where he thought he had a perfect spot to trap and destroy Richard’s army - but it didn’t work, and Richard defeated him.

So, there were storms and massacres but they took place before the march to Arsuf. Richard was on land with the army during the entire march and even participated in some of the smaller skirmishes that took place before the battle.

Sources:

John Gillingham, Richard I (Yale University Press, 1999)

Sidney Painter, "The Third Crusade: Richard the Lionhearted and Philip Augustus", in R.L. Wolff and H.W. Hazard, eds., A History of the Crusades, vol. II: The Later Crusades, 1189-1311 (University of Wisconsin Press, 1969)

The Chronicle of the Third Crusade: The Itinerarium Peregrinorum et Gesta Regis Ricardi, trans. Helen J. Nicholson (Ashgate, 1997)

Baha' al-Din ibn Shaddad, The Rare and Excellent History of Saladin*, al-Nawadir al-Sultaniyya wa'l-Mahasin al-Yusufiyya*, trans. D. S. Richards (Ashgate, 2002)