people can✀t afford to buy expensive things( )the prices of daily goods going up.

这是语法上的问题the prices of daily goods going up不是一个句子（注意going不是go原形），而是一个名词性从句．
如果选as for since虽然意思到位，但是要跟一个完整的句子the prices of daily goods go up而with后跟名词性从句the prices of daily goods going up

所以（people can't afford to buy expensive things with the prices of daily goods going up.人们不能承受购买昂贵的东西随着日用品的价格going up上升）
或者
（people can't afford to buy expensive things as/for/since the prices of daily goods go up.人们不能承受购买昂贵的东西，因为日用品的价格going up上升）

不理解可以补充，我会尽力解答

随着日用品的价格上涨，人们买不起昂贵的物品。

答案为C with是介词 +ing形式。

A。 as也是随着的意思，是连词，后面是一个完整的句子。
B。 for表原因，也是连词
D since自从，时间连词