In the head section of the master page you need to define a tag prefix in the style of:
<%@ Register Tagprefix="Yourwebpart" Namespace="yournamespace" Assembly="yourassemblyname, Version=18.104.22.168, Culture=neutral, PublicKeyToken=71e9bce111e9429c" %>
Also don't forget to change the PublicKeyToken and version number.
In the location where you want to place the webpart all you need to do insert:
<Youwebpart:YourClassNameOfTheWebPart id="mycustomwebpart" runat="server"/>
This is all because a web part is nothing more than a custom web control. If you have additional properties you can specify that too:
<Youwebpart:YourClassNameOfTheWebPart id="mycustomwebpart" YourPropertyName="PropertyValue" runat="server"/>
To add an image only to the start page you can also use a delegate control on the master page. For this you need to add to the master page a content place holder:
<asp:ContentPlaceHolder id="ShowOnlyMyImage" runat="server"/%gt;
The id "ShowOnlyMyImage" is only for reference you can change it to any id you like.
In the page layout you like to use for the home page you need define:
<asp:Content ID="MyAdditionlImage" ContentPlaceHolderID="ShowOnlyMyImage" runat="server">
ContentPlaceHolderID and ID here are also used only for reference. The ContentPlaceHolderID must match the ID in the MasterPage. For the ID you can use any unique name.