Can I check messages sent to a specific mobile number?

Yes. Here's how:

• Go to Reports > Detailed Reports
• Insert the full internationally formatted number in the number field (use the country dropdown if you need to change the country)
• Click Apply filters

Any messages to and from that number will be displayed in the log tables. 

If you can't find the data you are looking for don't forget to check the date range of the report as it may not be capturing messages sent to the desired contact.