There are 2 main meanings to phrasal verbs with across. The first is similar to a preposition of movement with an action verb + across to mean to get from one side to another (as in on a road for example). The more abstract meaning of phrasal verbs with across is to find by chance or to discover accidentally.
come across / happen across / stumble across : meaning to find or discover without looking intentionally. Example: 1. I came across some old photos when I was cleaning up. 2. The workmen happened across an old Roman villa when they were digging the new road. 3. The scientists stumbled across a new method of making anti-biotics. Of these 3 phrasal verbs meaning to find by chance, come across is the most common. Other phrasal verbs with a synonymous or similar meaning: come upon, stumble upon, run across.
MOVEMENT & DIRECTION
cut across – Meaning: to take a short cut. Example: We cut across the city using the side roads as there was less traffic.
get across – Meaning: to arrive to the other side. Example: The river was too high so we had to get across by boat.
run / walk / drive / ride / take across – Meaning: to move to the other side in the manner stated by the verb. Example: We ran across the bridge.
come across – Meaning: to change support. Example: The team were delighted when he came across to their side (i.e. joined their team).
get across – Meaning: to make an idea or thought understood by someone else. Example: Many students struggle with writing and can’t get their ideas across.
put across – Meaning: to explain or suggest or make an idea known. Example: Many ideas were put across during the meeting. Note the difference between get across and put across. Get across means that the other person understands while put means that it is said or stated (it’s not clear if the other person has understood).
come across – Meaning: to give someone an impression. Example: If you use your phone during the lesson, you’ll come across as not being interested.
keep across – Meaning: to stay informed. Example: Although she’s 90 years old, she keeps across of the news.