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Hilchos Brachos 52 (page 159)
מתחילת סימן נב עד אמצע סעיף א ומכל מקום

Omitting Parts of Pesukei D'zimra
Which Passages May be Skipped
Circumstances in Which Passages May be Skipped

Omitting parts of pesukei d'zimra
Someone who comes to shul after the minyan has started may omit some parts of the tefillos in order to say Shemoneh Esrei together with the tzibur. When very little time is available, he should try to say at least the berachos listed below and Baruch Sheamar, Ashrei and Yishtabach before beginning Birchos Kriyas Shema. If there is even less time, he may recite just the berachos of Al Netilas Yadaim, E-lokai, Neshama and Birchos HaTorah before beginning Birchos Kriyas Shema . He may omit Baruch Sheamar, Ashrei and Yishtabach. If time does not even allow for the proper recital of Shema, it is preferable for him to daven alone without catching up to the tzibur. Likewise, many poskim hold that if a person doesn't have time for Baruch Sheamar, Ashrei and Yishtabach, he should daven alone.
( סעיף א וס"ק א, ב, ו ו־ז; וראה ביה"ל לקמן סימן נג, סעיף א, ד"ה אין)
Which passages may be skipped
If there is only enough time for part of pesukei d'zimra, he should include passages in the following order of importance: 1 Ashrei, 2) Hallelu E-l B'kodsho, 3) Hallelu es Hashem, 4) the rest of the Hallelukah's, 5) the first half of Vay'vareich Dovid (until l'shem tifartecha), 6) the first half of Hodu (until v'hu rachum) and the pasuk " Hashem hoshia."  If after skipping he realizes that he has time to say the omitted sections, it appears from the Mishna Berurah that he should not return and recite those passages.
( סעיף א וס"ק ד; ביאורים ומוספים דרשו, 2)

Circumstances in which passages may be skipped
The exemption from reciting the full pesukei d'zimra applies only if there is no possibility of davening with a later minyan or if there is a chance of missing the zemanim for Shema or Shemoneh Esrei.  According to some poskim, a person who will miss davening with the tzibur but will be able to daven Shemoneh Esrei along with the chazan may also skip to catch up to the chazan. Other poskim are unsure whether this is permitted. It is questionable whether this exemption applies to women.
( ביאורים ומוספים דרשו, 1 ו־3)

  • A person should recite pesukei d'zimra in an unhurried manner, like someone who is counting his money, with care not to swallow or skip words.
  • The daily recitation of mizmor l'sodah corresponds to the Korban Todah. Accordingly, this mizmor is recited only on weekdays when the korban can be offered.
  • The verse of poseiach es yodecha must be said with attention to the meaning of the words. If it was recited without proper intention, this verse and the rest of the verses until the end of the paragraph must be repeated (even if the problem was realized after davening).

  • Reciting Birchos Hashachar at night before going to sleep

  • Reciting E-lokai, Neshama after sleeping during the day

  • Responding to kaddish or kedusha in the middle of Yishtabach