What's the specific use in carrying a pump?
Yes, people riding with a frame pump or a mini frame pump will use this in conjunction with a repair-kit (spare tube, patch kit, tire levers). 

Most if not all recreational cyclists carry some method of a repair kit because flats happen and it doesn't have to mean the end of your ride. Generally with practice a reasonable time to fix a flat is 5 minutes. Those 5 minutes spent fixing a flat will be much faster than calling an Uber or your SO for a ride, or even resorting to public transit. Better yet spend those five minutes changing a flat and then go finish your ride! 

You don't need a lot of bike tools—you, just need the right tools. A frame pump is one of those tools... 

A frame pump can be seen as a one time cost to get air into a deflated tire. Whereas others use a CO2 cartridge to inflate a tire, but over time a few bucks for one cartridge adds up. A pump and a CO2 cartridge have their advantages and disadvantages, but in the end the result is the same—fix a flat and be on your merry way!

Some satire on the matter can be found here.