@Darlene. You may wish to start a new thread. There are several professionals here willing to help. Yes you can fill a small gap with setting mortar depending on the mortars limitations. Some form of backing or reinforcement will be needed. As I stated above always go directly to manufacturers for installation instructions. Yes showers have a high degree of difficulty with a lot of details that must be followed. If you choose to proceed please purchase a TCNA Handbook. A $35 investment well worth it. You must use caution when seeking advise nowadays. I have found YouTube will have more bad info then good. Again go to the manufacturers videos! Unfortunately Handy Huber with his 1 project experience has resorted to name calling while providing you with bad advice. "then you have to run the cement board vertically from bottom to top." FACT Cement board is installed horizontally with staggered seams. ANSI 108.11 page 57 A-4.8.5.1.4 Cement board recommendations such as "new piece so that it's snug" "nice and tight." FACT The manufacturer instructions state leave a gap. https://www.usg.com/content/dam/USG_Marketing_Communications/united_states/product_promotional_materials/finished_assets/usg-durock-cement-board-with-edgeguard-installation-guide-en-CB237EG.pdf "And most importantly make sure when you get to tiling that you use trim tile ALL around the window frame! It's a crucial extra layer of protection against the constantly wet conditions and protecting the seams." FACT Tile has nothing to do with waterproofing. If you can't run the shower without tile it is not constructed properly. Many feel as I that showers are not a DIY project. Can it be done? Certainly. It's all in the details. Good luck with your project.

Can you post a picture to provide a bigger view of the window? In your close-up that appears to be some odd looking interior trim, like it was part of a window retrofit gone bad. If you just want to fill the gap, you can buy foam filler/backer rod (sold in various diameters) at the store, stuff the cavity with the foam backer rod, and then use paintable caulk over it to fully seal the gap. If white caulk is close to your paint color, you might not even have to paint it. You can also buy window foam sealer in cans, but I caution you about how hard it is to get off if you want to change anything, or even just to clean up your hands and tools. I would not use it for this type of job. Bruce

Legally, you are required to make the request for repair in writing. Include some pictures to make the situation clear. Request a response within 10 days. Send a written follow up that states that should this safety hazard not be repaired that you will be forced to have it done and deduct the cost of that from the rent that you owe. That’s the legal approach. If you’re getting a deal on rent, and don’t want to cause too many waves, you still need to document the issue and bring the condition of the property to his notice. For all defects. That protects you should he decide that he wants to blame the window falling out and cutting someone on you. Or the rot that goes down into the wall that the water damage has caused. He could blame that on you because you “never told him” about the issue. You need to notify him, even if he won’t fix it. A responsible landlord would want the opportunity to keep up his property.

Yes there should something to cover that area because if your tiles do not cover the the window will not look good and the finishing will not be neat and proper.