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I have heard that it's (1) meaningless and (2) a more playful version of ㄹ까. Not considering pronunciation, it also seems to be somewhat similar to the Japanese ending ~かな.
Examples
Also, what is its connection to ~려나?
Edit. Another example is "편파뉴스랄까나 편중된 뉴스가 많이 있기는 하지만" where it seems to mean something similar to "like".
I disagree that it's meaningless. To be meaningless, it could be removed from the sentence and the meaning should stay the same. It is not.
The link to the Japanese ending ~かな is https://namu.wiki/w/%EB%B2%88%EC%97%AD%EC%B2%B4%20%EB%AC%B8%EC%9E%A5/%EC%9D%BC%EB%B3%B8%EC%96%B4#s-4.3. I don't have any input on the similarity or dissimilarity to this Japanese ending.
However, the children's song in the first quoted box in the link is something I am familiar with. The meaning ~ㄹ까나 carries to me is like speaking aloud a question I have for myself. For example, the following same question can be said with different intonation as indicated by the arrow to be used for different purposes.
"점심이나 먹으러 갈까?↗" Question to others
"점심이나 먹으러 갈까.~→" Self-talking
"점심이나 먹으러 갈까나.~~→" Playful self-talking
Likewise,
고기를 잡으러 바다로 갈까나
고기를 잡으러 강으로 갈까나
conveys cheerful vocalization of one or more children whether to go to the sea or river to catch fish. If the ending 까 was used, it really doesn't give the same sense of amusing oneself or each other with this question essentially one is giving to oneself. I don't associate 까나 with sincere questions directed to others.
It is basically meaningless(or ignorable) or playful form. But sometimes can have trivial meanings.
할까나 ca be used in various situations.
When speaker pretty sure or (partially) agree with specific fact or opinion. Or speaker is not sure about the matter or fact.
The speaker is aware of the topic, but he or she wants to deliberately pretend to be ignorant on the matter. (Though listener knows exact fact or their opinion)
The common concept of the ㄹ할까나 is what speaker wants to say is not the fact or matter itself but opinion (sometimes it implies asking for listener's opinion, not objection) with intention of avoiding criticism or objections.
—랄까나 is too.(~라고 할까나)
This is really tricky to a foreigner. Also, it is not a standard form of Korean. As I know, it is not listed on Korean dictionary.
It can be understood as irresponsible. So need to be avoided on public. Many Koreans use this word only in SNS or with friends.
Regarding ~려나(~ㄹ려나) It is prediction form with asking
오늘 비가 올려나? → It might be rain today.(?)
까나,and questioning까나?are different. For the former the closest meaning in English I can think of isor.이걸까나, 저걸까나...,그럴까나, 아니면 저럴까나...And까나?is used when teasing someone.어쩔까나?그럴까나?누굴까나?