That's called a "sucker," which is just a term for a tomato branch which grows from between the main stem and a compound leaf. Its appearance actually doesn't have anything to do with the broken leaf branch: that's just coincidence. You'll eventually see the same thing happening where each leaf branch joins the main stem.

There's an old belief that suckers rob the main plant of strength (hence the name). Not true! They're just the tomato's normal way of growing branches (after all, not every type of plant has the same pattern of growth). After growing a few leaf branches of its own, the sucker will produce flowers, and from then on will behave just like the main stem -- including growing suckers of its own!

If you keep the sucker, your plant will be shorter and wider, which is a good thing, since otherwise most tomato plants will grow taller than you can reach (though some varieties are naturally short, and climate plays a role too). The leaf cover will be heavier, which means the fruit will have more protection from potentially-damaging summer sun.

How many suckers to keep is up to you. Debates on this subject can get every bit as hot as discussions of religion or the issue of PC vs. Mac. Many people don't prune at all. It depends on how much space you have and what sort of supports you're using. [Generally speaking, more stems = more tomatoes; however, to some extent more tomatoes = smaller tomatoes, so people trying to grow tomatoes of record size often limit stems and the number of fruit.]

I don't grow tomatoes in containers, but I gather that a 10 gal. bucket should be fine.

Please note that tomatoes are either indeterminate (sometimes called vines) or determinate (sometimes called bushes).

Indeterminates (indets) essentially keep growing forever, until killed by frost, disease, accident, etc.

A determinate's stems, however, grow to a certain length (i.e., a certain number of leaves and flower clusters) and then stop. Each stem terminates in a cluster of flowers. The End (except for the fruit ripening). So it's important not to prune suckers from det tomatoes.

By the way, you can cut off a sucker and root it in damp potting mix to make a new plant. Tomatoes are amazing!

That's pretty interesting. Thanks for the info! I already pruned a lot of suckers of that plant because I heard it is The Thing To Do, but it didn't occur to me that the 'branch' that is growing could be a sucker. I will seriously consider hanging on to it.

I like to prune off lower suckers but leave upper ones to encourage more of the fruit to be at a convenient height. Everyone has their own approach.