The base level (round shield) Carthaginian cav are good enough on very hard/very hard. You need to hit two at a time, you will take few losses because the infantry will rout about the time they both make contact. And getting the long shield cav helps, as well as various upgrades.
Use the Balearic slingers, that will force the enemy to come to you...and your cav can slaughter them wholesale.
The Carthaginian starting position is incredibly weak though. When playing other factions the Romans take Carthage out in short order. It will bet worse if CA fixes the reluctance of the AI to blockade ports. Part of the problem is that everyone gangs up on Carthage. So you have 3 AI Roman factions, plus Numidia, Spain, and Gaul going after them on the ground, plus the Senate boats will attack them at sea.
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