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You have a copy error. This means that Joomla! can't write to the directory where the extension's files are stored. In this case you need to enable the FTP layer on your site, as described in https://www.akeebabackup.com/documentation/troubleshooter/abinstallation.html

Moreover, you need a writeable temp directory for Joomla! to handle uploaded files. Now, I am going to tell you something that will probably make you cringe, but bear with me. Start by creating a file named htaccess.txt with the following content:

order deny,allow

deny from all

and upload it in your site's tmp directory. IMPORTANT! Pay attention on what you're doing in this step. Do not upload the file in the root of your site!
Give the htaccess.txt file you just uploaded 0644 permissions.
Now rename it to .htaccess (the DOT in front of the name is important).
Go up one folder. Now give the tmp folder 0777 permissions. Please give these permissions only to the folder, not its contents.
You can now proceed with the update.

At this point I would recommend using a better host with a better grasp on security. If you don't understand why, please read the Security Information chapter in the Akeeba Backup User's Guide. As you see, having Apache run under a different user as the one who owns the files of your site leads to security issues. A good host will either be using virtualisation or a solution like FastCGI Process Manager (php-fpm) or suPHP to have the Apache process run under the same user as your files' owner, greatly improving the security.

OK, hold it. You are applying a "security advice" without having the faintest clue of what it does. Please read the "Security information" chapter in Akeeba Backup User's Guide (yes, I know, I did say the Akeeba Backup's documentation). In there I have written a concise introduction on users, groups, ownership and permissions in UNIX/Linux systems for regular humans. Read it a couple of times (I've only seen one person getting it from the first read – it is a lot of information!) and you'll understand what these permissions do.

If you followed my suggestion, you now know what the problem is. Unwriteable files. You were asking Joomla! to write to unwriteable files. Now look at what I replied to you five days ago:

You have a copy error. This means that Joomla! can't write to the directory where the extension's files are stored.

Now you see that I was spot on :)

Sorry, I missed your reply :(

The "best" settings are 755 for folders and 644 for files. However, this doesn't mean that your folders will be writeable by PHP. It all comes down to who owns the file, under which user PHP runs and the permissions.

Rule of thumb: always set folders to 755 and files to 644. If you can't install a software you will need to do two things.

1. Making your temporary directory writeable
Give the tmp directory 777 permissions. Since this is a security threat we have to mitigate the risk. It's really easy! Create a new file called htaccess.txt with Notepad and put the following contents into it:

order deny,allow
deny from all
allow from none

Upload it to your site's tmp directory. Give it 644 or 444 permissions. Rename it to .htaccess (dot htaccess – the dot in front is important!).

2. Enable the FTP layer in Joomla!
Go to Global Configuration and set up the FTP options in Joomla!.

Please note that this information is readily available in https://www.akeebabackup.com/documentation/troubleshooter/abinstallation.html