Pompey's wife Cornelia was not with him at - nor on the run with him after - the Battle of Pharsalus. He rendezvoused with his wife Cornelia and his son Sextus Pompeius on the island of Mytilene, and from there went to Egypt, where he was assassinated as soon as he stepped ashore.
Pompey, Cicero, Cato, and Brutus are shown meeting in Pompey's camp after the battle, but according to history this did not happen. In fact, Caesar captured the camp and found that it had been decorated for a party to celebrate Pompey's victory.
When Pompey is explaining the battle of Pharsalus to Lucius Vorenus he says that his cavalry was repulsed by a single cohort of Caesar's reserve infantry. According to history, Caesar's reserves in that battle totaled nine cohorts.
Brutus and Cicero are shown surrendering together to Caesar after the Battle of Pharsalus. In reality, only Brutus surrendered immediately after the battle, since Cicero wasn't even present at the battle. Cicero remained with Cato and other Pompeian loyalists until news of Pompey's defeat and subsequent murder reached them, causing Cicero to immediately judge that the war was lost. He refused the offer made to him by Cato of supreme command of their remaining army, enraging Pompey's son Cnaeus to the point where he almost killed Cicero before Cato intervened and stopped him. Cicero returned to Italy and surrendered to Mark Antony, but was told that only Caesar could officially pardon him. Cicero then met with Caesar as the general was leaving Brundisium, and he was instantly pardoned.
Caesar is shown coldly ignoring Cicero after he and Brutus surrendered to him after the Battle of Pharsalus. In reality, Caesar and Cicero had a strong mutual respect for each other and corresponded via letters for years, even when Caesar was fighting in Gaul. When Cicero surrendered to Caesar he was given a warm welcome, instantly pardoned and urged to go back to Rome to continue playing a part in Senatorial life. Contrary to how the series depicts the scene, Caesar was actually very happy to have Cicero back on his side. Before Cicero fled Italy and joined Pompey's forces, Caesar had even repeatedly asked him to stay in Italy and remain on his side, since he respected and valued Cicero's counsel so much.