I'm afraid I have got to go as I have got a meeting.
I don't want to go, but I have a prior engagement.
Sometimes when you are at an event or with friends you may have to leave early as you have plans to meet someone already. This can be called a 'prior engagement'.
A 'prior engagement' is a arrangement to meet someone at a particular time and place.
例えば他の人と会う約束をしているときなど、他の友人と一緒でも、その場を早めに出なくてはならないこともありますよね。このような先約のことを英語では、"prior engagement"と言います。
"A prior engagement"は、特定の日時に、特定の場所で人と会う約束をしている、という意味です。
"I hate to leave, but I need to go."
This is a basic sentence that you can add to in order to explain what you hate to leave but why you need to.
Hate - when you are sad, disappointed, or upset that you have to leave a friend or an event.
Examples:
"I hate to leave you guys, but I need to pick up my son from school."
"I hate to leave early, but I need to take my car to the mechanic."
"I hate to leave the party, but I have an important meeting in the morning."
"I hate to leave you here alone, but if I do not leave now I will be late for work."
I hate to leave, but I need to go.
（帰りたくないですが、帰らなければなりません。）
これは基本的な文章で、あなたが帰りたくはないですが、帰らなければならない理由を追加することができます。
Hate - 友達やイベントを残して去らなければならず、悲しく、がっかりしていて、またいらいらしています。
例：
I hate to leave you guys, but I need to pick up my son from school.
（あなた達をおいていくことは残念ですが、息子を学校に迎えに行かなければなりません。）
I hate to leave early, but I need to take my car to the mechanic.
（早く出なければいけないのは残念ですが、車を修理にもって行かなければなりません。）
I hate to leave the party, but I have an important meeting in the morning.
（パーティーを退席するのは残念ですが、朝に重要な会議があります。）
I hate to leave you here alone, but if I do not leave now I will be late for work.
（ここにあなたを1人で残すのは残念ですが、仕事に遅れるので出なければなりません。）
The expression 'duty calls' means that you have to be elsewhere in order to attend to some necessary work. 'A pressing engagement' means that you have, perhaps, another appointment, or must be in another location for some reason. The third example sentence is informal and is self-explanatory.
Unfortunately something has come up and I need to go.
I wish I could stay, but I've another engagement to attend.
Unfortunately something has come up and I need to go. - by using unfortunately you communicate that something unexpectedly or not planned has come up and needs you attention, so you need to leave.
I wish I could stay, but I've another engagement to attend. - here you express your desire to stay, but you've already made plans with someone else and need to leave immediately.
Unfortunately something has come up and I need to go.（残念ですが、ちょっと用事ができてしまって、もう行かないといけません）
- "Unfortunately"（残念ながら）を使って、急用ができて、もう行かないといけないと伝えています。
I wish I could stay, but I've another engagement to attend.（もっと長くいられたらいいんですが、別の約束があるので）
- ここでは「もっと長くいたいけど、別の約束があるのでもう行かないといけない」と伝えています。
Apologizing because you have to leave quickly or earlier because you have run out of time is more polite.
"I really must get going", is more urgent, abruptly finishing your visit/cpnversation because you have to leave urgently.
Saying good-bye for now means leaves an opening to resume the conversation/visit at a later time.
「申し訳ないがもう行かないといけない」と謝罪を入れた方が丁寧です。
"I really must get going" は切迫しています。「もう行かないと」と言って、唐突に話を終わらせています。
"I must say 'good-bye' for now"（今日のところはもう行かないと）はその後会話を再開する余地を残しています。
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