。
I asked them to send me another ticket.
Literal
I [topic-は] them [target-に] ticket [object-を] one-more-counter send-give-[purpose-ように] requested.
Two constructions deserve attention. ~てくれる ('to do for me') marks the action as a benefit received by the speaker — without it, the sentence would lack the implication that the ticket is being sent for the speaker's benefit. ~ように+頼む is the indirect-request pattern: 'to ask in such a way that X.' もう一枚 uses 枚, the counter for thin flat objects (sheets of paper, tickets, plates).