Well, I have watched many people drink espresso with and without sugar over the years, so I feel compelled to jump in and say that, for me,
"I often drink an espresso with sugar" doesn't have the same meaning as "I often add sugar to my espressos."

"I often drink an espresso with sugar." To me, this means @TOMO_o drinks espresso often, and every time he drinks a cup of espresso, he puts sugar in it.

"I often add sugar to my espressos" To me, this means that sometimes the speaker doesn't add sugar to his espresso. (If he added it every time, then he would say "I always add" sugar)

In my experience, people do not tend to add sugar sometimes but not others when they order or make espresso.