What is the name of this manga with a girl pretending to be a mermaid?
The manga you are looking for is:
Satou Kashi no Dangan wa Uchinukenai 
(Sugar Candy Bullets Can’t Pierce Anything).



The following summaries are included on the website:


  When a strange new girl by the name of Umino Mokuzu transfers in from
  Tokyo and declares that she is a "Mermaid", she quickly becomes the
  center of attention. Noticing that Nagisa is the only classmate that
  seems to have no interest in her, Umino quickly decides that they
  should become "precious friends". Despite Nagisa's objections, Umino
  begins to follow her wherever she goes.


and


  Umino's rockstar father is shown to be aloof towards her at best and
  emotionally abusive at worst, once leaving her behind in a grocery
  store parking lot simply because a defective shopping cart made him
  angry and that's only the tip of the iceberg.