I've come to enjoy using the word 微睡む lol. 微笑む(ほほえむ) and 微睡む(まどろむ) have a very easy to understand construction when you look at the kanji.
For 微笑む, it just means to smile, but if you look at the two individual kanji 微 and 笑, you see that they mean "delicate/minuteness" and "laugh" repesectively. Put those bad boys together and BOOM, you get smile (or the delicate laugh). You tend to see 微 in other words like 微生物(microbe) or 微細(micro, minute) so you instantly know that when you see a word like 微笑む( or 微睡む), you know it's going to be somewhat related to something small/minute.
For 微睡む, just do they same thing and BOOM, you get to doze off(or the delicate sleep).
I don't think I've heard or seen the word 微睡む used, spoken or written, outside of my kanken study materials lmao. I would imagine it's probably because there are a few other ways to express "dozing off" or "taking a nap" like:
転た寝する(うたたね)- doze, nap, snooze
ごろ寝する(ごろね)- falling asleep (in one's clothes), lying down to nap, crashing out
昼寝する(ひるね)- nap (afternoon), siesta
仮眠する•仮眠を取る(かみん)- nap, doze, to catch some Z's
Etc!
Now you can throw 微睡む right in there with the other ones making the list a little bit longer haha
by Domotenno