It occurs to me that the image of Hazel as “Mother Time” is an allusion to the character “Kate” the old charwoman in Joyce’s Finnegans Wake. Kate appears with slop pail and broom, and like Time, is always cleaning up the mess after disasters, picking out various detritus which she has been throwing on her garbage heap since the beginning of time. The word “time” is frequently found somewhere near her. I am not aware of the image of "Mother Time" with slop pail and broom occurring anywhere else except in Finnegans Wake. Here is a description of her from FW (I like the fact that the image of “footprints” is also contained here):
“Kate Strong, the old scavenger-widow who knew the city during those filthy times paints for us a dreary, glowing, vivid picture…she let down, as scavengers who will be scavengers must, her filth dump…And she chose this place because all over it were the complicated traceries of the past: fossil footprints, book marks, elbow dints, etc. What subtler time-place could anyone as for the hiding of a love letter, than then when ructions ended and here where race began?” (79-80)
Also, "HAZEL SHADE" suggests a near-anagram of "NADEZHDA" ("hope"), that is, there is "some small (part of) hope" within her name that helps her father with his existential crisis. And yet, her father despairs of her ever becoming what he thinks she should be. Hope cannot exist without Time; both cease for Hazel as a "watchman, Father Time" and his "uneasy dog" (Time and Death) come on the scene. Then the only hope is hope for eternity.