Some phrasal verbs describe a problem that appears or that you find by chance. Each one keeps its own particle, and the particle carries the meaning. Drop it and the sentence stops meaning "appear".
When a problem appears
come up /kʌm ʌp/ — A problem came up before the event at Senville Park.
crop up /krɑp ʌp/ — Technical problems cropped up during the online lesson.
come across /kʌm əˈkrɔːs/ — Mei came across an error in the old report.
Not ""A problem came before the event." (particle dropped, so this just means "arrived earlier")" — say ""A problem came up before the event."". The particle makes the meaning: come up = appear, but come alone does not.
Try it: While reading the old report, Mei ___ a mistake she had never noticed before. (came across / came up / cropped up) Answer: came across — Come across means find something by chance, which fits noticing a mistake while reading. Come up and crop up mean a problem appears on its own, and they do not take an object like "a mistake" this way, so they do not fit here.
The same three, now for any problem
You have already met work out, figure out and sort out for problems at work. They are not tied to work — they fit any problem at all. The difference worth holding on to: work out and figure out mean you find the answer by thinking, while sort out means you deal with a mess successfully and put things back in order. All three keep out.
work out /wɜːrk aʊt/ — Sofia worked out a way to reduce the waiting time.
figure out /ˈfɪɡjər aʊt/ — Yasmin figured out what caused the mistake.
sort out /sɔːrt aʊt/ — Farid sorted out the registration problem.
Not ""Diego worked why the file would not open." (no particle, so this is not standard English)" — say ""Diego worked out why the file would not open."". You work out an answer, and the particle out is part of the verb.
Try it: The registration system kept crashing, but Farid finally ___ the problem so students could sign up. (sorted out / came up / cropped up) Answer: sorted out — Sort out means deal with a problem successfully, which matches fixing the registration so it works. Come up and crop up describe a problem appearing, not being solved, which is the opposite of what the sentence needs.
Try it: Hiroshi studied the code for an hour and finally ___ why the app kept freezing. (figured out / came across / cropped up) Answer: figured out — Figure out means understand or find the answer after thinking, which fits studying the code to find the cause. Came across is finding something by chance rather than after effort, and cropped up means a problem appears, so neither describes reaching an answer.
Meeting a problem you did not expect
Run into means to meet a problem or a person unexpectedly, and it always takes an object. Break down means to stop working, and it is used for machines, vehicles and negotiations. Both keep their particle, which carries the meaning.
run into /ˌrʌn ˈɪntuː/ — Yasmin ran into a problem with the booking system.
break down /ˌbreɪk ˈdaʊn/ — The delivery van broke down outside Mosh Market.
Not ""The van broke outside Mosh Market." (particle dropped, so this means it was damaged)" — say ""The van broke down outside Mosh Market."". Break down = stop working. "Break" alone means something is damaged or in pieces.
Try it: Diego was printing the reports when the office printer ___ and stopped completely. (broke down / ran into / came up) Answer: broke down — Break down means to stop working, which is what a printer that stops completely has done. "Run into" needs an object, because you run into a problem, and "come up" describes a problem appearing rather than a machine failing.
Try it: While setting up the new accounts, Farid ___ a problem nobody had predicted. (ran into / cropped up / came up) Answer: ran into — Run into means to meet a problem unexpectedly, and it takes an object, so you run into a problem. "Crop up" and "come up" describe a problem appearing on its own and take no object, so neither can be followed by "a problem" here.
Handling a problem rather than solving it
These two verbs are about handling a problem rather than finishing it. Deal with means to take action on something that needs attention. Cope with means to manage a difficult situation without being overwhelmed by it. Compare sort out, which means the problem is finished.
deal with /ˈdiːl wɪð/ — Hiroshi dealt with the complaints from the morning shift.
cope with /ˈkoʊp wɪð/ — Lani coped with the extra work while her colleague was away.
Not ""Hiroshi dealt the complaints." (the preposition is missing)" — say ""Hiroshi dealt with the complaints."". Deal with keeps with before the problem. Without it, "deal" means to hand out cards.
Try it: The clinic was short of staff all winter, but Matilda ___ the pressure by planning each day carefully. (coped with / cropped up / came across) Answer: coped with — Cope with means to manage a difficult situation without being overwhelmed, which fits working through a whole winter of pressure. "Crop up" means a problem appears and "come across" means to find something by chance, so neither describes managing a situation.
Try it: Two customers were waiting to complain, so Wei asked Mei to ___ them while he finished the order. (deal with / come up / work out) Answer: deal with — Deal with means to take action on something that needs attention, which fits attending to waiting customers. "Come up" describes a problem appearing on its own, and "work out" means to find the answer to a problem, so neither works with people who need serving.