At most, it would apply to only one roll of Healing Spirit and only if the DM allows it
Rule wording is ambiguous
The ambiguity here results from the unwritten assumption in the Moon Sickle's description that healing spells generally heal immediately when they are cast. And if all healing spells did that, there would be no issue.
However, Healing Spirit does not heal immediately upon being cast (and, in fact, may not ever heal at all) which then causes a potential split in how you read the following phrase:
When you cast a spell that restores hit points ...
You can read that either as:
- When you cast a spell that restores hit points [immediately as part of its casting] ...
- When you cast a spell that restores hit points [at any point the spell is active] ...
Neither one is strictly backed up or negated by rules text that I can find, so that leaves it up to the DM at your table to decide.
Neither interpretation gives a bonus on every roll of Healing Spirit
Interestingly, regardless of which interpretation your DM goes with, the bonus healing would not apply to each roll of Healing Spirit.
In interpretation 1, the bonus would not apply at all since Healing Spirit does not heal immediately upon casting.
In interpretation 2, the bonus would apply once since it says
When you cast a spell that restores hit points, you can roll a d4 and add the number rolled to the amount of hit points restored
In other words, you cast a healing spell and it increases the total number of hit points restored per casting by 1d4. Thus Healing Spirit would only get 1 additional 1d4 of healing over the entire course of the spell.
My ruling
My personal thought and ruling at my table would likely be to keep things simple and interpret it as healing spells that heal immediately upon casting (interpretation 1). This has the dual benefit of being simple and of not increasing the power of an already powerful (some may even argue over-powered) Healing Spirit.
But even if your DM decides to read it as interpretation 2, it would only apply to one roll of Healing Spirit.