You have to edit /templates/esmobile/index.php in line 150-157 you find this:
<?php if (Esmobile::easyblogExists()) { ?>
<div class="es-mobile-menu__item">
<a href="<?php echo EBR::_('index.php?option=com_easyblog' . EBR::getItemId('latest'));?>">
<i class="fa fa-file-text-o"></i>
<span><?php echo JText::_('COM_EASYSOCIAL_TOOLBAR_BLOG');?></span>
</a>
</div>
<?php } ?>
You can remove the code or replace it by something like this:
<div class="es-mobile-menu__item">
<a href="URL WITH SLASH IN FRONT">
<i class="FONTAWESOME CODE"></i>
<span><?php echo JText::_('TITLE ON BUTTON');?></span>
</a>
</div>
URL WITH SLASH IN FRONT= Put your URL the button should go to. Let's say google.com/supercool is the link you need then put /supercool.
FONTAWESOME CODE= Pick and FontAwesome icon you like from here:
http://fontawesome.io/icons/ and insert the code of the icon you see on the linked page after clicking the wanted icon.
TITLE ON BUTTON= This will appear on your site as the title of the button.