Yes, there's a quest to repair your broken starter weapon for every Elithian race! In all cases, you'll need to complete the Apex mission in order to gain access to the quest - this is so you can't access the powerful repaired version of your weapon before reaching an appropriate power level yourself. For an Avikan character, complete that mission, then take your broken spear to the Starfarer's Refuge: there will be a new quest waiting for you there.

If you ever play as a Droden or Trinkian character, the same rule applies: complete the Apex mission, then take your broken starter weapon with you to a set location. For the Droden, the quest is also found in the Starfarer's Refuge. For Trinks, you need to take your weapon to any Trinkian settlement instead.

How do you get to the Refuge? I saw another discussion, and I completed Barron's Keep, but the quest won't show up. I only have the quest "Report to Commander Zaheed". I'm REALLY confused.

The way to it is randomly found on desert and grassland planets.
Just have to keep looking.
You will either find a small settlement or a larger one, with a teleporter to it.

I forgot, what is the droden's starting weapon?

ƁЄƦƓЄ ƓƠƠƁԼƖƝ の投稿を引用：

How do you get to the Refuge? I saw another discussion, and I completed Barron's Keep, but the quest won't show up. I only have the quest "Report to Commander Zaheed". I'm REALLY confused.

The Starfarer's Refuge can be found by accessing a teleporter that links to it. You can find these teleporters in any Avikan settlement or NPC ship, or you can craft it yourself if you're playing as an Avikan.

Nameless Nate.j2 の投稿を引用：

I forgot, what is the droden's starting weapon?

The Droden start out with a weapon called the Broken Longsword.