What is incumbent upon me if I break an oath that I specified to a certain date to stop eating particular foods for the purpose of losing weight, and in which I vowed to pay a large sum of money that I do not possess if I break the oath?

If you break your oath, you are required to offer an expiation for breaking an oath (kaffarat yameen). The expiation for breaking an oath is to feed ten poor people, or to clothe them, or to free a slave. If you are unable to do any of these, then you must fast for three days.

The vow that the questioner made regarding eating food is considered a vow concerning a permissible act (nadhir al-mubah), which, according to the majority of scholars, is not binding, and breaking it does not necessitate an expiation. However, according to those scholars who validate it (such as the Hanbalis), it is not obligatory to fulfill it, and one has the choice between performing the act or refraining from it, along with an expiation for breaking an oath.

Since the questioner linked their vow to breaking their oath, the issue is connected to the ruling on fulfilling an oath: If fulfilling the oath is permissible, then it is a vow concerning a permissible act, and its ruling is as stated above. If fulfilling the oath is recommended, then it is a binding vow, and an expiation for breaking an oath is required for it.

The ruling on fulfilling an oath varies depending on its purpose. If you swear to abstain from a disliked matter (such as pleasant things), then it is sunnah to break the oath. The preference varies according to people's circumstances and intentions. However, if the oath is concerning a permissible act with no specific purpose, then it is better to fulfill the oath to glorify the name of Allah.

Therefore, it is more cautious and preferable for the questioner, if they break their oath, to offer expiation for their vow in addition to the expiation for breaking the oath, because fulfilling their oath might be recommended, in which case two expiations would be required.