This answer focuses more on addressing the issue in future semesters. If you have room to innovate, it may help to focus on your desired course outcomes and let assignment policy (and even the assignments) flow from them.
With a set of outcomes in mind, and a sequence of major assignments that (I hope) all play a role in reaching them, I knew I wanted to orient all of my policy around getting students to work hard, complete the full assignment track, and learn from their mistakes (both in coursework, and time-management/responsibility).
This orientation led me to a few specific conclusions:
- Getting students to learn from mistakes means: making sure they have room to make them, providing clear feedback that they messed up, and expecting them to rectify it.
- Unless I receive notice of a university-verified excused absence, I apply a diminishing daily late-work penalty that ensures it's worthwhile to turn assignments in on the due date, but leaves enough points on the table that it's always still worth completing late assignments.
- Break major assignments down into a few parts to dilute the impact of an occasional discretionary absence or missed due date.
- Require corrections on all major assignments.
- Aim for a well-distributed workload with weekly deadlines and steadily-increasing assignment weights. This communicates my expectations (and puts procrastinators on notice) with clear grade feedback while the stakes are low.
A relaxed attendance or late-work policy isn't directly compatible with some kinds of work, but I think it's a good nudge to re-examine assumptions. As long as you still make appropriate full-credit accommodations for people with officially-documented excused absences, I think it's fine to have an office-hours make-up option (i.e., same exam for half credit, much harder essay exam, etc.)