お負（ま）けに is a phrase which often functions as a conjunction between the (A) and (B) parts of a sentence in order to show that (B) is ‘in addition to (A)’.
This structure stems from the common use of the noun お負（ま）け which indicates a ‘discount’ or ‘freebie’ that is added in addition to the base of (A). Despite お負（ま）け itself primarily being used for positive things お負（ま）けに may convey ‘in addition’, ‘on top of that’, or ‘what’s more’ regardless of whether the original topic is a good thing or not. However, (A) and (B) will always either be both good things, or both bad things, never a mixture of both.
Being a combination of お負（ま）け and に, this phrase will appear either at the beginning of a new sentence, or at the beginning of a second clause of a 2-part sentence.

• 今日（きょう）は仕事（しごと）に遅刻（ちこく）して部長（ぶちょう）に怒（おこ）られたし、おまけに取引先（とりひきさき）の人（ひと）も怒（おこ）らせちゃったから、今日（きょう）は最悪（さいあく）の日（ひ）だったよ。
  Today was one of the worst days because I got yelled at by my boss for being late to work, and to make it worse, I upsetted one of our clients.
• 昨日（きのう）は彼氏（かれし）に美味（おい）しいご飯（はん）をご馳走（ちそう）してもらって、おまけにプレゼントまでもらった。
  Yesterday, my boyfriend treated me to a delicious dinner and on top of that he even gave me a gift.
• パソコンが全然（ぜんぜん）立（た）ち上（あ）がらない。おまけにスマホの充電（じゅうでん）がないから仕事（しごと）が全然（ぜんぜん）できない。
  My computer won’t turn on. To make matters worse, my smartphone is out of battery, so I can’t work at all.