According to Iwanami Bunko's recent collection of haiku by Nagai Kafū (荷風俳句集, ed. Katō Ikuya 加藤郁乎), this is a hauta written as a jingle for Matsuya, back when it was Matsuya Gofukuten ("Matsuya Japanese-Clothing Store"):

松屋呉服店広告用端唄

〽湯上りのおくれ毛なづる涼風も一雨ごとに身にしみていつか今年の秋袷。過ぎし花火に見覚えのぬしが羽織の紋どころそつと崩してわたしのと比翼に染めて知らぬ顔。やがて浮名のたて縞に末はめでたう子持じま。川といふ字に寐てみたいといふて朝夕神いのり。

The cool wind that strokes the hair on the nape of my neck sinks deeper into my skin with every shower of rain, and now I find myself in a lined autumn kimono. A man I know from some past night of fireworks has quietly dyed out the crest on his haori and replaced it with a design combining his and mine, and pretends not to know me. My name no longer worth a pin/stripes, but in the end with baby making three/-fold stripes. Longing to share a futon as a family, I pray morning and night.

I believe that we are to understand this as the tale of a courtesan who has fallen in love with a man whose social position prevents him from openly acknowledging her at present, but who nevertheless makes gestures (the shared crest design, etc.) that give her hope that he will make an honest woman of her one day.

I haven't made any effort to reproduce the poetic effect (note the 5/7/5/7 structure) but I did make a rudimentary attempt to recreate the key puns at the end:

yagate ukina no/ tatejima ni/ sue wa medetō/ komochijima
My name no longer worth a pin/stripes, but in the end with baby making three/-fold stripes.

Tatejima literally means "vertical stripes", and here it's overlaid on ukina o no ta[tsu], "rumors start." Komochijima means "child-having stripes"; it's another vertical stripe design that pairs each thick stripe with a thin one (go here and search for "子持縞"). Here it's overlaid on komochi, i.e. the concept of literally having a child, starting a family. (Note also the use of sue!)

Other items of linguistic interest:

• The source term for "design combining his and mine" is hiyoku 比翼, short for hiyoku mon 比翼紋 (where mon means "crest"). Hiyoku comes from hiyoku no tori, which corresponds nicely to "lovebirds." It's originally from a line in Bai Juyi 白居易's Song of Eternal Sorrow (長恨歌):
  
  

  在天願作比翼鳥 / 在地願為連理枝
  "If we are in the heavens we will be like birds flying with wingtips together/ If we are on earth we will be intertwined branches of a tree." (Chris Kern's translation)

• The source term for "share a futon as a family" is kawa to iu ji ni nete 川といふ字に寐て. This is a well-known expression that I think first appeared in the Edo period. It literally refers to a family of three sleeping with the child between the two parents, like the character 川 (kawa, river), and metaphorically connotes contented family life proceeding as it should.