Since you’ve already tried the header padding, size, tagline, and font adjustments, I’d check the actual mobile CSS next. In particular, I’d look for overflow, fixed heights, or width restrictions on the title/header container.
I’d also temporarily disable any custom CSS and test with plugins disabled one at a time. If the problem only happens on this particular site, that should help separate a theme issue from something specific to the site’s CSS or plugins.
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