Is expiation incumbent upon someone who gave money to a husband for the treatment of his wife, who later died after a month, and the questioner suspects that the death was due to the husband's negligence?

Your speed in a crowded place at night was a mistake. Consult the hospital to determine the cause of death. If it is proven that the accident was the cause, then you owe expiation for accidental killing: freeing a believing slave, or fasting for two consecutive months, along with repentance, and paying the woman's blood money to her family (this falls upon the 'Aqilah, unless they pardon it). If it is not proven, then the original state is absolution of responsibility, but for the sake of your faith, it is more cautious to perform the expiation. The evidence for this is Allah's Almighty saying: "And never is it for a believer to kill a believer except by mistake. And whoever kills a believer by mistake – then the freeing of a believing slave and a compensation payment presented to his family, unless they give [it in charity]..." [An-Nisa: 92].