. . . will they cover it . . . ?

No: they will blame you for "working unsafely" after denying you the resources--time, tools, and training--to have possibly avoided injury.

. . . Randy Austin (the employee) fell while mopping a restroom floor at the Kroger where he worked in Mesquite, Texas. The employee was a self-described floor clean-up person whose supervisor instructed him to clean up two oil-like spills, one in the womens bathroom and one in the mens bathroom. Krogers safety handbook indicated that employees who clean these types of spills should do so using a Spill Magic system that involves a powdery absorbent product, a broom, and a dustpan. This system likely reduces the chances of a slip-and-fall by about 25% (according to the Kroger handbook). But, this system was not available at the store the day the employee was instructed to clean the spills, thus the employee attempted to clean the spills with a mop. After successfully cleaning the womens bathroom, the employee began to carefully clean the mens bathroom. The employee successfully cleaned about 30% of the spill before he slipped and fell in the remaining amount of liquid, resulting in a fractured femur, a dislocated hip, and a nine-month hospital stay. During the employees hospital stay, he underwent six surgeries, leaving his left leg two inches shorter than his right. -- http://texastechlawreview.org/supreme-court-of-texas-update-austin-v-kroger-texas-l-p/

https://www.jw.com/tag/randy-austin-v-kroger-texas/