Translation:Kateřina asked me to wait for her.
The direct or indirect objects are mostly terms used for the English language and its usability for Czech is limited. In Czech various verbs take various kind of objects using various cases and prepositions. Požádat (ask, request), mostly uses accusative for the person you ask something from and if you ask "for something" it is "o něco" ("o" + accusative).
You have to learn the cases and prepositions for each verb individually. At least those that do not just use a simple accusative for one single object.
I suggest to review the introductory Tips and notes about Czech orthography and pronunciation. Namely those about the effect of ě after certain consonants.
These combinations are pronounced as:
bě - /bje/
pě - /pje/
vě - /vje/
mě - /mňe/
So that mně and mě are pronounced exactly the same.
And even more different:
dě - /ďe/
tě - /ťe/
ně - /ňe/
The reason is historical because /mje/ changed to /mňe/ many centuries ago.
It might also be good to review the effect of "i" in "di", "ti", "ni".
I am native AmE, and "Kateřina requested..." would be acceptable as an alternative to "Kateřina asked..." in this context, though "asked" is probably used more often. (Note also that the Czech sentence is in the past tense, not the present.)
I suspect that some translations are just missing; the exercise has not been edited for a few years. Sentences like, "Kateřina asked/requested THAT I wait for her" also are not accepted, but with these there is a question of how well they fit the Czech original.
[Clarification: My reference to "a question of how well they fit the Czech original" was intended to convey uncertainty on my part, since I am native AmE, rather than reluctance to accept the "asked that..." alternative on the part of the native Czech speakers on the team to add it.]
Thank you for the information. I understand your reluctance to translate aby… by “that,” since aby… is conjugated. On the other hand the English infinitive construct (“asked me to wait”) is possible only when the subject of the subclause is identical to the object of požádat. I suppose (can't be sure though) that the given sentence would work without mě, too, and then you'd have no choice but translating with “that:” “Kateřina asked that I wait for her.” (She need not have talked directly to me.)
In fact my Czech dictionary translates aby by “that,” in the following form: aby (~ch, ~s, ~, ~chom, ~ste, ~) = that (I, you, he / she, we, you, they). So maybe it would be a good idea to accept translations with “that.”
My dictionary defines požádání as a neuter noun meaning “request” or “demand.” So “on request” would be “na požádání.”
At Czech bus stops you can frequently find a sign saying “na znamení,” literally “upon sign,” which can also be translated as “on demand.” It means that you have to give the bus driver a sign, otherwise they might pass without stopping.