Trials of Ascendancy: I'm a bit confused

2nd act, normal difficulty.

I've got two relics/coins. I did first ascendacy, failed miserably at the second one (some random explosion killed me). Can I redo the second one or is it lost forever?

In general second Act tends to be a bit confusing.

Edit: I did this Baba(whatever) boss again, and she dropped nothing.

Quest given ascendancy chances are not singular attempts.
Aka you can try them again and again, until you succeed.

I personally ran Temple of Chaos trial 4-5 times. 2-3 times i died, 1 time i crashed and succeded at last. Inscribed ultimatum remains in device, so you can stop trial and start it again from zero.
And from people i play with, they ran Sekhema Trials multiple times.

Okay, but how can I do it again? I can't see the option.

Baba at the trials entrance gives another coin.
Just talk to her

Tried every single conversations with her, no coin.