I am using sp 2013. When I click on "Sites" in the top bar I go to this page:
http://mijn.companyName.com/persoonlijk/hajjk1/Social/Sites.aspx#IsEditing_WPQ4=1
Is it possible to provision a custom webpart on this page for each new mysite?

I am using sp 2013. When I click on "Sites" in the top bar I go to this page:
http://mijn.companyName.com/persoonlijk/hajjk1/Social/Sites.aspx#IsEditing_WPQ4=1
Is it possible to provision a custom webpart on this page for each new mysite?

Yes, it is possible. The best approach would be a feature with a feature receiver that you would activate (programmatically or via PowerShell script every time you create a MySite).
Take a look at the following pseudo code which should be in the future receiver:
var web = SPContext.Current.Web;
var mySitePage = web.GetPage(web, SPUtility.ConcatUrls(web.Url, _NEWPAGE_PATH_));
if (mySitePage != null && mySitePage .Exists)
{
using (SPLimitedWebPartManager webPartManager = mySitePage.GetLimitedWebPartManager(PersonalizationScope.Shared))
{
var yourCustomWebPart = new YourCustomWebPart();
webPartManager.AddWebPart(yourCustomWebPart, "_WEBPARTZONE_LOCATION_", 1);
webPartManager.SaveChanges(yourCustomWebPart);
}
}
_NEWPAGE_PATH_ should be your current MySite page file name.
_WEBPARTZONE_LOCATION_ should be your web part zone location.
If you need any clarification, please let me know.