I'm not sure you need the $k => $p where you have it. It should work with this:
<?php foreach($page->children as $child): ?>
<div class="mask">
<img src="<?php echo $child->images->first()->url; ?>" />
<h2><?php echo $child->headline; ?></h2>
<?php echo $child->body; ?>
<a href="<?php echo $child->url; ?>" class="info"><?php echo $child->title; ?></a>
</div>
<?php endforeach; ?>
If I'm understanding you correctly, this should get you most of the way to where you want to go. I'm not a php expert and I haven't tested this, but I think that's right. Let us know if that works.
thistimj's post in How do I get image from children was marked as the answer