My sentence:
people in poor countries use this strategy to make 1 dollar per day. I know it's not a lot of money but they can call it extra income (not Americans or Japanese).
can say: they can do call it extra income?
My sentence:
people in poor countries use this strategy to make 1 dollar per day. I know it's not a lot of money but they can call it extra income (not Americans or Japanese).
can say: they can do call it extra income?
The "do" form is not simply emphatic, it has the sense of "contrary to your expectation".
I do like dogs!
Doesn't mean "I like dogs very much". It means "Although you think I don't like dogs, you are wrong!"
It is ungrammatical to use "do" in this structure with modals like can, will or might.
I do play tennis (ok). I can play tennis (ok). I can do play tennis (not okay)
In speech the emphasis you want would be made with intonation. This can be indicated in writing by italics:
I can play tennis.
or with adverbs
I really can play tennis.
In your example (which is a little unclear) you can add emphasis with italics.
I know it's not a lot of money but they can call it extra income.
The implication is that "although you think that they can't call it extra income, you are wrong."
In practice I think you need to think about what you want to say. I think "... but it is extra income" is clear and focuses the sentence on what the money is, not what it is called.