I had a similar question recently as we had a brit milah in shul and the grandfather distributed a prayer for personal requests to be said at the time of the brit. I was surprised this was allowed and looked it up afterwards.
Turns out, in a nutshell, that, beyond the traditional nusach, requests for spiritual matters are allowed while requests for material issues should be avoided.
R Yirmiyohu Kaganoff writes here for instance
In several places, the Gemara mentions that one may not pray for
individual needs on Shabbos (e.g., Taanis 19a; Bava Basra 91a;
Yerushalmi, Shabbos, 15:3). At least two reasons are quoted for this
prohibition. Some sources include it under what the Navi Yeshaya
(58:13) commanded when he declared, Vechibadto mei’asos derachecha,
mimetzo cheftzecha vedabeir davar, "You shall honor the Shabbos by not
performing your own matters, seeking out your own needs and speaking
of them" (Vayikra Rabbah 34:16; Rashba, Shabbos 113a). This
proscription is usually simply called dabeir davar.
Others prohibit praying for personal requests on Shabbos because it
violates one’s oneg Shabbos. Praying for personal needs causes one to
focus on what troubles him, which leads a person to be sorrowful (see
Rambam, Hilchos Shabbos 30:12 and Ran, Shabbos, Rif page 5b). Shabbos
is to be a day of joy.
[R Kaganoff then goes into a long discussion why the traditional
nusach contains personal requests, see original for details]
The Mishnah Berurah (288:22) permits praying on Shabbos for spiritual
help or for any other request that is not a result of difficult
circumstances. It seems that this should be permitted for two reasons:
(1) one should not pray on Shabbos about one’s own needs, but
spiritual needs are Hashem’s realm and (2) most people do not become
saddened regarding their spiritual failings and "troubles."
R Moishe Dovid Lebovits writes similarly here (see there for sources)
Chazal forbade one to ask for his personal needs on Shabbos. Some say
this is learned from the din that on Shabbos one has to act
differently from the rest of the week. Others say the reason is
because it is not an oneg for Shabbos if one asks for his personal
needs. The Ran says the reason is because asking for personal needs on
Shabbos may cause a person to cry on Shabbos which is not allowed. The
Korbon Eida says the reason is because of tirdas halev.
Many poskim say the whole issur is to ask regarding a tzara one
already has, and one is permitted to request from Hashem that a
certain tzara should not come. Some say something which has a set
nusach can be said on Shabbos even if it contains personal requests.
Although some poskim prohibit asking for personal ruchniyus needs on
Shabbos, many poskim say it is permitted to do so. The minhag is in
accordance with this second opinion. The reason is because asking
for ruchniyus needs does not conflict with any of the reasons