Is there any word for 10 days in English?

Solution 1:

No single word is current usage for a period of ten days. A week covers a period of seven days, a fortnight covers a period of 14 days, but there is nothing in-between which can be considered current usage.

Solution 2:

As others suggested, there is no English word that could mean "10 days". But the term "10 day" is broadly used for weather forecast and report. You can use

• 10-day report for the first 10 days of a month
• 10-day report for the middle 10 days of a month
• 10-day report for the last 10 days of a month

Based on context, you can use the first/middle/last part of a month.

Example: MANDATED 10-DAY REPORT

[Maryland Board of Physicians]