The most common and most likely answer to this crossword clue is the 7 letter word ROUGHER
I've seen this crossword clue in Independent
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More clues leading to the result ROUGHER
Buddy explains!

The straight part
We know that "with greater severity" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case with greater severity = rougher.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does ROUGHER mean?
ROUGHER adjective- Not smooth; uneven.
- Approximate; hasty or careless; not finished.
- Turbulent.
- Difficult; trying.
- Crude; unrefined
- Violent; not careful or subtle
- Loud and hoarse; offensive to the ear; harsh; grating.
- Not polished; uncut; said of a gem.
- Harsh-tasting.
- Somewhat ill; sick
- Unwell due to alcohol; hungover
- A worker who shapes something to prepare it for a finishing operation.
- A piece of woollen cloth prepared for fulling.