In this context, "speak into" translates as "address" because it implies communicating directly and purposefully with the culture to deliver a prophetic message or guidance.

Example sentences:
1. The leader addressed the congregation, reminding them to speak into society's injustices with love and compassion.
2. As a writer, she aims to address pressing issues in her novels, speaking into the hearts and minds of her readers.
3. The activist group plans to speak into political debates, offering their perspective and urging for change.

looks like bad English and really makes very little sense. And you have given no surrounding sentences that would help devipher it.

Only 3 Ghits for "speak into culture" and adding "prophetically" just one!

We use "speak into " for speaking INTO a device like a microphone or phone but otherwise it's "speak ON"=comment, or "speak ABOUT/TO" = address

At first, I thought they maight have wanted to say "talk into" = persuade or convince, as many non natives get "talk" and "speak" confused


I suggest foretell, predict for the "prophetically" part

The church was told to speak into culture prophetically >
The church was told to comment on culture in the future >
The church was told to address the issue of culture in the future>
The church was told to predict/foretell what would happen to culture

but more context IS needed

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Note added at 1 hr (2024-01-11 17:07:31 GMT)
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Sorry! typo 2nd line: should be "decipher"

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Note added at 6 hrs (2024-01-11 22:54:27 GMT)
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OK so I think I understand it better now having read some more.

This seems to be a particular meaning in religion to do with getting people to understand where the church is working/what it's doing/what are its intentions or aims while the church in turn tries to understand where/how people are within their own culture and lives and thus come to a better mutual understanding

https://www.stgeorgesepsom.org.nz/wp-content/uploads/2019/11...


"Because the more we engage with culture, the more we can understand culture.
The more we understand culture, the more we can speak into culture.
And the more we can speak into culture
the more we can communicate our FAITH, IN CULTURE."
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Anyway, the word I'd use now would probably be COMMUNICATE (with people on their own level and within their own cultural norms)

English term (edited): The church was told to speak into culture prophetically

Clue:
"I told The New York Times that the church of Jesus Christ and culture have always been at odds, and that we need to understand that the church was never called to accommodate culture," he says. "The church was called to prophetically speak into culture."
https://www.charismanews.com/marketplace/86772-megachurch-pa...

IOW what this pastor meant:

the church's "calling" was to infuse its prophecies into the outside culture. The church is meant to do the "speaking" and (the outside) culture to do the "listening", NOT the other way round.

BTW, this logic makes perfect sense from the viewpoint of this church, as it would make sense from the viewpoint of any religion / cult / ideology that all seek to expand their influence.