how to patch minor leaks at cast iron sewer pipe's joint
There is no such thing as a "minor" leak in cast iron. If it is leaking, it means major corrosion is occurring. If you were to open the joint and look in, you would see the inside of the pipe is completely rotted.

You could put a liner inside the pipe and pump it full of some kind of sealant, but it would be a lot of work to do that and it would be temporary fix because the rot would continue at the interface between the liner and the pipe.

The correct way to do it is to first remove the short, corroded cast iron piece and then grind/chisel/sand the larger flanged opening. You need at least 1.5 inches depth of clean, smooth cast iron. Then you put in the PVC pipe and caulk it, which means stuff it with oakum and melted lead. You have to have at least 1" depth of lead all around the joint. Doing this requires a plumber who is experienced with caulking. It requires special tools called "caulking irons". You may find some plumbers who recommend using a "donut" adapter. Don't do it. Those cheapo things are worthless garbage used by hack plumbers. If your plumber even mentions the word "donut", fire him and get a plumber who is not a moron. Caulking is the way to go. These are pictures of what it should look like when done:



By the way, that rubber adapter with hose clamps you have on there is a complete POS typical of the worst sort of hack plumber trying to finish a job in 5 minutes. You should get rid of that thing when you do the repair and replace it with proper, solid PVC joints.