When attempting to update a self-hosted WordPress site, I’m seeing “ASN1 unknown message digest algorithm”

@Kika is right that you have an older, insecure, version of OpenSSL.

This sounds like:
https://wordpress.org/support/topic/getting-download-failed-error0d0890a1asn1-encoding-routinesfunc137reason

I wrote a small bit of code to skip the SSL test when trying to access the download server. It is not a good idea to use this code but will get you out of a pinch.

<?php
/*
Plugin Name: Skip SSL Verify
Plugin URI: http://www.damiencarbery.com
Description: Skip SSL verify in curl downloads - fixes: Download failed. error:0D0890A1:asn1 encoding routines:func(137):reason(161).
Author: Damien Carbery
Version: 0.1
*/

function ssv_skip_ssl_verify($ssl_verify) {
    return false;
}
add_filter('https_ssl_verify', 'ssv_skip_ssl_verify');
add_filter('https_local_ssl_verify', 'ssv_skip_ssl_verify');