Mrs. Johansen rose quickly from the chair by the window where she'd been sitting. Mrs. Rosen, Ellen's mother, was there, too, in the opposite chair. They'd been having coffee together, as they did many afternoons. Of course it wasn't really coffee, though the mothers still called it that: "having coffee." There had been no real coffee in Copenhagen since the beginning of the Nazi occupation.
Not even any real tea. The mothers sipped at hot water flavored with herbs.

What does "at" mean or imply here as in "sipped at"? If we used without "at"?

Question

Mrs. Johansen rose quickly from the chair by the window where she'd been sitting. Mrs. Rosen, Ellen's mother, was there, too, in the opposite chair. They'd been having coffee together, as they did many afternoons. Of course it wasn't really coffee, though the mothers still called it that: "having coffee." There had been no real coffee in Copenhagen since the beginning of the Nazi occupation.
Not even any real tea. The mothers sipped at hot water flavored with herbs.

What does "at" mean or imply here as in "sipped at"? If we used without "at"?

Answer

Here, 'sipped at the hot water' implies a disinterest in drinking it. As the passage states, it's not real tea or coffee but it instead represents a comforting ritual distracting her from the reality of the war around her.

We also say to 'poke at her food' to communicate that a person isn't really interested in eating.

Normally we would just say 'she sipped her tea,' but including 'at' implies a disinterest or perhaps that her mind is preoccupied with other things.