Ruby, 99 89 # Ruby, 157 bytes

->y,m{(a=(Date.new(y)..Date.new(y,m,-1)).reject{|d|d.sunday?|d.saturday?}[-1].strftime("%A %-d"))+((b=a[-1])>?3?"th":b>?2?"rd":b>?1?"nd":"st")}
require'time'

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+68 Bytes to fix restricted output

Never thought Ruby had ranges also for dates!

Explanation

Anonymous function tacking year and month as numbers.

We pass the [First day of year .. last day of month(-1)] range and reject Sundays and Saturdays.

We return the last element.