English Translation
Tom asked his girlfriend to pardon him for forgetting her birthday.
There are two problems in this Japanese sentence. First, it sounds like a robotic word-for-word translation. Second, 請う (pronounced as こう) should be replaced with 乞う (also pronounced as こう). Here are my alternative translations:
恋人の誕生日を失念したトムは、平謝りした。（こいびとのたんじょうびをしつねんしたトムは、ひらあやまりした）
彼女の誕生日をうっかり忘れてしまったトムは、平謝りした。（かのじょのたんじょうびをうっかりわすれてしまったトムは、ひらあやまりした）
ガールフレンド is a loan word from the English word “girlfriend”, but we seldom use it in a daily life. 彼女（かのじょ）is often translated as “she/her” in English, but the primary meaning of 彼女 in the modern common use is actually “a girlfriend”. When we refer to a female third person, we rarely say 彼女 but use her name (メアリーさん etc.) instead. 彼女 is completely interchangeable with 恋人 if you want to clarify that the female third person is Tom’s girlfriend.
失念する is a frequent expression you all had better memorize and use in the real world. 失念する means “unintentionally forget something to do”. The connotation is “ooooopsie, sorry” in a polite way. In a casual conversation, you may want to replace 失念する with うっかり忘れる or ど忘れする.
平謝り is another expression to memorize. 謝る（あやまる）as a verb means “to apologize”. Conjugated with 平（ひら）, it means “to kneel down and beg for forgiveness”. 平らな（たいらな）as an adjective means “flat”, so it illustrates the beggar bowing his head to the same level as the ground (i.e. his head and body are as flat as the ground).
許しを請う is not usually used in the context of “to deeply apologize”, but “to get permission”. So, the person who 許しを請う didn’t do anything wrong, but he merely claims or requests for permission. For example, you ask your supervisor for a leave of absence from your workplace. It’s just a procedure. 請求書 (せいきゅうしょ) with the Kanji 請 is “a billing” (request for payment). The sender of 請求書 didn’t do anything wrong. It’s not begging.
You need to replace 請う with 乞う (also pronounced as こう) in this “apologizing to his girlfriend” context. 乞う means “to beg”. 乞食 means “a beggar”. 乞うご期待！ is a frequent term you see at the end of a TV drama, meaning “to be continued, and please keep your expectation high until the next episode”.
However, 許しを乞う in this apologizing to Tom’s girlfriend context is still unnatural. 平謝りする is much better.