Is Syr about to marry and have a heir?, judging by Avernus, I predict that people marry at a younger age than in our world.
Could Syr be sent off world, preferably permanently, to live a normal life or to complete her training, at the very least, she will have better chances of surviving the death world.
What about collecting genetic samples from Syr and her father, for a emergency case scenario? For cloning, surrogate pregnancies and other options. Are artificial incubator tanks available?
What about using diviner psykers to help predict deaths or act as a alarm system for important people or Bio-psykers to sense hostile life forms to better survive in Avernus.
What about establishing more defenses for cities like a electrified fence, mines and autoflamethrowers to scare of predators or some sort of radar to detect predators, sterelising enviroments to reduce plaques, health care plans. Bunkers for civillians to hide in or airships out of predator's reach. Chemical warfare for other nasties of Avernus. Bombs hidden in human shaped objects to be used against huge monsters.
Why can't psykers just retake tests if they just fail instead of being executed. Its not as if they have limited time, or if needed why not use drugs or devices to suppress their powers and let them retake the exam.
Why not train a animal thats good at detecting larger animals in Avernus, it could decrease the casuality rate.
How bad is a chaos cultist, for example a father invoke chaos to save his ill daughter, a guardsman who knows hes going to die, its not as if they lost all free will and turns into a psychotic mass murderering monster who wants the world to burn overnight, if chaos cultists can still remain hidden then they could still feel emotions and think like rational humans and be persuaded to stop. Do they all have to die? Does a abomination cultist look and act like a normal human being or do they look like a horror movie monster?
What about culture to improve morale, manga, anime, amusement parks, water resorts, internet, video games, dating sites. All of these could be powered by normal power plants and would improve the lives of civillians. What about hosting the show ' Ciphias Cain'.
Studying how psychic powers work. And interact with human brains , can't corpses be used to boost a psyker powers, don't their abilities come from a brain.
Why not have void shield floating platforms as a means of defense against enemy artillery.
Create remote controlled scouting and detecting drones who can scout out dangers and threats for armies before they began a expedition. What about flying vessels for scouting?
The sirens they are intelligent right? Do they look like mermaids, eat human flesh, reproduce with human men?
Is diplomacy a option?