I wonder if this might be [*Spear of Heaven*][spear] (1994) by Judith Tarr.

[![Cover of "Spear of Heaven" showing Daruya holding her daughter Kimeri against the backdrop of a Worldgate; Daruya is holding her right hand in front of her, with the glowing sun sigil visible on its palm][1]][1]

Daruya, having produced Kimeri, heir to the line, wishes to leave the palace of the emperor, her grandfather Estarion, despite the potential for calamity that the unexplained collapse of a Worldgate suggests.

> "Daruya," he said a little wearily, "no one ever forgets that you have given the empire an heir. It's still a remarkable scandal."
> 
> "What, that I wouldn't name her father, let alone marry him? Believe me, Grandfather, you wouldn't want him playing consort to my imperial majesty, when I come to it, which pray god and goddess won't be for long years yet. He's a beautiful, brilliant political idiot."

A burning brand on their hands:

> She snatched the warrior from the board and flung it at him. He caught it in a hand that flashed gold—like her own, like her daughter's. Like that of every heir to the throne of the Sun. They carried gold in their right hands like a burning brand, born there—set there by the god, the priests said. She did not know. It burned, that she knew, and worse, the more she fought it.

The brand (called the *Kasar*) on the hands of the ruling line - the Emperor Starion, Daruya and Kimeri - is the sign of the Sun god (this is made explicit in an earlier book) and the brand and its power are identified as the sun:

> She freed the right from her cloak and unfolded her fingers from her palm.  The sun in it shone dazzling.

The black-clad bodyguard is Chakan, one of the Olenyai; he is not mute, but traditionally silent in front of those not of the household he is bound to.

> A shadow leaned on it, regarding her with golden eyes. He was faceless else—
veiled, hooded, black-robed from head to foot. He inspired no fear in her at all, and no surprise. “Chakan,” she said.  “What are you doing up at this hour?”

  [1]: https://i.sstatic.net/Bhj1y.jpg
  [spear]: http://www.isfdb.org/cgi-bin/title.cgi?14459