Would you mind sending this letter for me?
My dictionary states that 郵便物 means “mail, postal matter, postal items”
So I think it should be “Would you please send this mail for me?”
I suppose the sentence could refer to multiple letter, or maybe a package or two.
この郵便物を出す is most likely to be interpreted as “to drop this letter or postcard into a post box”. There is no connotation of plurality in this sentence structure. The verb 出す implies that it won’t be an over-the-counter transaction between the customer and the delivery operator, but the letter/postcard will be collected from an unmanned post box.
If it’s a package, we usually say 荷物を送る (にもつをおくる) rather than 郵便物を出す.
The business of 郵便 was dominated by Japan Post (郵便局), which used to be a state-owned enterprise and privatized in 2000s. Japan Post is still the solo owner of red post boxes across the country. On the other hand, several private companies have been performing very well in the package delivery business. Notable market leaders are Yamato (ヤマト運輸) and Sagawa (佐川急便) although Japan Post is also engaged in the package delivery business. The business of Yamato and Sagawa is not called 郵便事業 because of the historical regulation. That’s probably why I interpret 郵便物 as a letter or a postcard.