Yes, Hallowed Moonlight exiles enchantments put into the battlefield with Starfield of Nyx if you control 4 or more enchantments.
Hallowed Moonlight creates a replacement effect that lasts until end of turn. If Hallowed Moonlight wasn't in effect, the enchantment would enter the battlefield as a creature as long as you control 4 other enchantments. It doesn't matter what combination of types it was outside of the battlefield, all that matters with that types it would enter the battlefield.
614.1. Some continuous effects are replacement effects. Like prevention effects (see rule 615), replacement effects apply continuously as events happen—they aren’t locked in ahead of time. Such effects watch for a particular event that would happen and completely or partially replace that event with a different event. They act like “shields” around whatever they’re affecting.
That's where Hallowed Moonlight steps in and exiles the card, because it never sees the card as anything other than an enchantment creature.
603.6b Continuous effects that modify characteristics of a permanent do so the moment the permanent is on the battlefield (and not before then). The permanent is never on the battlefield with its unmodified characteristics. Continuous effects don’t apply before the permanent is on the battlefield, however (see rule 603.6e).
Therefore, the enchantment would enter the battlefield as a creature, and Moonlight replaces that event with exiling that creature.
Note that there is the following rule:
614.12. Some replacement effects modify how a permanent enters the battlefield. (See rules 614.1c–d.) Such effects may come from the permanent itself if they affect only that permanent (as opposed to a general subset of permanents that includes it). They may also come from other sources. To determine which replacement effects apply and how they apply, check the characteristics of the permanent as it would exist on the battlefield, taking into account replacement effects that have already modified how it enters the battlefield (see rule 616.1), continuous effects generated by the resolution of spells or abilities that changed the permanent’s characteristics on the stack (see rule 400.7a), and continuous effects from the permanent’s own static abilities, but ignoring continuous effects from any other source that would affect it.
The last line would say that Starfield of Nyx wouldn't be considered for determing whether the Moonlight replacement effect applies to enchantments. However, the rule specifically only applies to how permanents enter the battlefield, not whether they enter it at all. It applies to cases with a wording as described in
614.1c Effects that read “[This permanent] enters the battlefield with . . . ,” “As [this permanent] enters the battlefield . . . ,” or “[This permanent] enters the battlefield as . . . ” are replacement effects.
614.1d Continuous effects that read “[This permanent] enters the battlefield . . .” or “[Objects] enter the battlefield . . .” are replacement effects.
This does not fit moonlight's wording, so 614.12 can't apply to Moonlight. There are also no other specific rules that would apply to Moonlight's replacement effect, so I have to assume that Moonlight has to consider type-changing effects such as from Starfield, just like for example ETB triggers have to consider it.