If one accidentally skipped part of Pesukei Dezimra, may he recite it between Yishtabach and Yotzer Or?
If one accidently skipped part of Pesukei Dezimra, it is permitted to recite the skipped paragraphs between Yishtabach and Yotzer Or. Thus, if one accidently skipped Az Yashir, and already recited the blessing of Yishtabach, he may recite Az Yashir between Yishtabach and Yotzer Or. Likewise, if one accidently skipped Nishmas on Shabbos or Yom Tov, and already said the prayer of Yishtabach as he is accustomed to do during the week, then it may be recited between Yishtabach and Yotzer Or. [The same applies if one purposely skipped part of Pesukei Dezimra in order to catch up with the Minyan, and then realized that there would anyways be a delay until Shemoneh Esrei. Likewise, if one did not recite one of the paragraphs with concentration, such as Ashrei and Poseiach Es Yadecha, then it may be repeated between Yishtabach and Yotzer Or.]
May one say Tehillim between Yishtabach and Yotzer Or?
No. One may not recite Tehillim or verbally learn Torah between Yishtabach and Yotzer Or. One may however learn from a Sefer in his thought.
 Birkeiy Yosef 281 in name of Michtam Ledavid 13 “One who forgot to say Nishmas and already said Yishtabach as he is accustomed during the week, is to say Nishmas after Yishtabach” brought in Shaareiy Teshuvah 281:1; Ben Ish Chaiy Toldos 2; Kaf Hachaim 281:7; Ateres Zekeinim 56 regarding certain Pesukim that “One may say these verses even between Yishtabach and Yotzer and need not worry of a Hefsek.”; Meorer Yisheinim 183; Piskeiy Teshuvos 54:3; See Admur 54:4 and other Poskim [Rama 54:2; M”A 54:3; Kol Bo 6; Sefer Hamachkim; Orchos Chaim] who all rule that if the Chazan made a Hefsek after Yishtabach he is to repeat some of the Pesukei Dezimra; See Admur 51:4; 53:3; 54:3 and other Poskim [Michaber and Rama 54:1; Tur in name of Seder Rav Amram Gaon; Kol Bo 4] who rule one may make an interval between Yishtabach and Yotzer Or for the sake of a Mitzvah; See Shevet Hakehasi 6:65; Kovetz Teshuvos 1:19
 The reason: As the prohibition against making an interval between Yishtabach and Yotzer Or is only regarding mundane matters, however to mention praise of Hashem which is similar to the content is certainly allowed. [Ateres Zekenim ibid]
 Piskeiy Teshuvos ibid
 Implication of all Poskim who question the recital of Shir Hamalos between Yishtabach and Yotzer, during Aseres Yimei Teshuvah [See Magen Avraham 54:2; M”B 54:4]; Igros Kodesh 8:99 regarding learning Chassidus that “certainly you are only thinking it and not verbalizing”; Yabia Omer 6:5 in name of Poskim; Piskeiy Teshuvos 54:3
Other opinions: It is implied from several Poskim that it is permitted to recite Tehillim in between Yishtabach and Yotzer Or, just as we permit saying Shir Hamalos Mimamakim between Yishtabach and Yotzer. [See Degul Merivava 54 who answers the question of the M”A ibid by stating that the prohibition of an interval is only applicable regarding matters irrelevant to prayer and not with regards to words of praise. So also answer the Aruch Hashulchan 54/2; Toras Chaim Sofer 54/2. This argument should likewise apply towards permitting saying Tehillim. This is in addition to the allowance recorded in Admur and Poskim ibid to make a Hefsek for the sake of a Mitzvah between Yishtabach and Yotzer and seemingly the same would apply to Tehillim and even learning Torah. However, see above that we explained that practically today we are stringent unlike this opinion.]
 Aruch Hashulchan 53:4 [however does not explicitly state that it must be done within one’s thought]; Igros Kodesh ibid; Beir Moshe 3:20; Yabia Omer ibid