Postgres HAVING ilike any wildcard in array_agg

I need a filter where the input value is similar to any value in an aggregate, using ilike and wildcards. My attempt at a query so far is:

SELECT p.id, p.name, array_agg(vo.name)
FROM product p
LEFT JOIN product_variation pv
ON p.id = pv.product_id
LEFT JOIN variation_option vo
ON pv.variation_option_id = vo.id
GROUP BY p.id
HAVING $1 ilike any(array_agg('%' || vo.name || '%'));

But it doesn't work. It gets even more complicated: ideally, I'd be able to input an array of strings, where any of them are similar to the aggregate. Any tips? I realize this kind of works, just not as I need it to. The input value should be able to be a partial match for any of the array elements, but it is the other way around. The wild card would need to be in the input value.