As we know, Adding list view web part is different from custom web part using powershell, what's more, there are also difference between adding web part to web part zone page and wiki pag. here is the method.

1. Add custom web part to wiki page:

Note: because of custom web part, we couldn't new the web part via new-object Microsoft.SharePoint.WebPartPages.XsltListViewWebPart, had to get the web part from the folder of the web part catalog. first, we should get the custom web part via name in the web part folder, second, read the custom via OpenBinaryStream() method and import the file to web part object.

And, because of add web part to wiki page, the wiki page didn't have web part zone, but it has the hide zone named "WPZ", after adding web part to the wiki page, we still couldn't see it, the reason is that the wiki page is reloadey by the html code, wo had to re-write the html:

<div class="ms-rtestate-read ms-rte-wpbox" contenteditable="false" style="float:left;width:30&%;min-width:300px;">

 <div class="ms-rtestate-notify  ms-rtestate-read $($lvwpGuid)" id="div_$($lvwpGuid)" unselectable="on"></div>

 <div id="$($lvwpGuid) " unselectable="on" style="display: none"></div>

</div>

so we just change the id of div, then we will get the result. 
Here is the function:

AddCustomWebPart http://localhost "SitePages/Home.aspx" "TrendingTagsWebPart_TrendingTags.webpart" "Trending Tags"

function AddCustomWebPart($siteCollectionUrl, $pageUrl, $webPartName, $title){

$site = new-object Microsoft.SharePoint.SPSite($siteCollectionUrl);

$web = $site.OpenWeb()

$defaultPage = $web.GetFile($pageUrl)

$item = $defaultPage.Item

#Create fancy GUID

$lvwpGuid = [System.Guid]::NewGuid().ToString()

$lvwpKey = "g_" + $lvwpGuid.Replace("-","_")

$errorMsg = ""  

[Microsoft.SharePoint.SPList]$wpList = $site.GetCatalog([Microsoft.SharePoint.SPListTemplateType]::WebPartCatalog)

[Microsoft.SharePoint.SPFolder]$wpFolder = $wpList.RootFolder

[Microsoft.SharePoint.SPFile]$wpFile = $wpFolder.Files[$webPartName]

[System.Xml.XmlReader]$xmlReader = [System.Xml.XmlReader]::Create($wpFile.OpenBinaryStream())

[Microsoft.SharePoint.WebPartPages.SPLimitedWebPartManager]$wpManager = $defaultPage.GetLimitedWebPartManager([System.Web.UI.WebControls.WebParts.PersonalizationScope]::Shared)   

$myCustomWP = $wpManager.ImportWebPart($xmlReader,[ref]$errorMsg)

$myCustomWP.ID = $lvwpKey

$myCustomWP.Title = $title

$wpManager.AddWebPart($myCustomWP, "WPZ", 0); 

# Add the HTML content and web part containers to the page.

 $wikiContent = @"

<div class="ms-rtestate-read ms-rte-wpbox" contenteditable="false" style="float:left;width:30&%;min-width:300px;">

 <div class="ms-rtestate-notify  ms-rtestate-read $($lvwpGuid)" id="div_$($lvwpGuid)" unselectable="on"></div>

 <div id="$($lvwpGuid) " unselectable="on" style="display: none"></div>

</div>

"@

#wiki content is stored in the field “Wiki Content”

$item["WikiField"] = $wikicontent

$item.Update()

$xmlReader.Close()

$web.Dispose()

$site.Dispose()

write-host "Done"

}

1. Add list view  web part to wiki page:

AddWebPartToWiki http://localhost "SitePages/Home.aspx" "Links" "LinkOne"

function AddWebPartToWiki($siteCollectionUrl, $pageUrl, $listName, $viewName){

$web = get-spweb $siteCollectionUrl

$list = $web.Lists[$listName]

$wpPage = $web.GetFile($pageUrl)

$item = $wpPage.Item

# Get the LimitedWebPartManager

$webpartmanager=$wpPage.GetLimitedWebPartManager([System.Web.UI.WebControls.WebParts.PersonalizationScope]::Shared)

#Create fancy GUID

$lvwpGuid = [System.Guid]::NewGuid().ToString()

$lvwpKey = "g_" + $lvwpGuid.Replace("-","_")

# Instantiate wp

$lvwp = new-object Microsoft.SharePoint.WebPartPages.XsltListViewWebPart

$lvwp.ID = $lvwpKey

$lvwp.WebID = $web.ID;

$lvwp.ChromeType = "TitleOnly";

$lvwp.Title = "Your Title";

#$lvwp.TitleUrl = "http://dev-sp";

$lvwp.Toolbar = "No Toolbar";

$lvwp.ListID = $list.ID;

$lvwp.ListName = $list.ID.ToString();

# Set the view

$lvwp.ViewGuid = $list.Views[$viewName].ID.ToString();

# Add the web part

$webpartmanager.AddWebPart($lvwp, "WPZ", 0);

# Add the HTML content and web part containers to the page.

 $wikiContent = @"

<div class="ms-rtestate-read ms-rte-wpbox" contenteditable="false" style="float:left;width:30%;min-width:300px;">

 <div class="ms-rtestate-notify  ms-rtestate-read $($lvwpGuid)" id="div_$($lvwpGuid)" unselectable="on"></div>

 <div id="$($lvwpGuid) " unselectable="on" style="display: none"></div>

</div>

"@

#wiki content is stored in the field “Wiki Content”

$item["WikiField"] += $wikicontent

$item.Update()

# Update the web

$web.Update();

$web.Dispose();

write-host "success"

}

More to Link: http://soufiane-benyoussef.blogspot.in/2011/09/add-custom-webpart-to-page-using-power.html