我觉得这里用冠词the或代词their更好，因为这个还款显然指的是和homes 有关的。我不知你是从哪里看到的不能用the的说法。

我倒是觉得这里的carry on的用法，比较过时了，现代英语已经不太用了，即使用也通常用carry on with. 例如：If at any time you feel you can't carry on with the payments, we will take back the machine and credit you with the sum you've paid up to that time.

我觉得carry out 的用法更多见些。1. Turkey wanted to carry out the repayments in a manner that would not affect its general economic situation. 2. But the defendant - appellant in accomplishing this result, never intended to carry out the payments for such physical changes to the structure and the loss of rent by the plaintiff - respondent .