<?xml version="1.0"?>
<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:ObjectManager/etc/config.xsd">
<preference for="MagentoCatalogWidgetBlockProductProductsList" type="VendorNamespaceBlockProductProductsList" />
</config>

<?php
namespace VendorNamespaceBlockProduct;

class ProductsList extends MagentoCatalogWidgetBlockProductProductsList
{
    protected $_httpContext;

    public function __construct(
        MagentoCatalogBlockProductContext $context,
        MagentoCatalogModelResourceModelProductCollectionFactory $productCollectionFactory,
        MagentoCatalogModelProductVisibility $catalogProductVisibility,
        MagentoFrameworkAppHttpContext $httpContext,
        MagentoRuleModelConditionSqlBuilder $sqlBuilder,
        MagentoCatalogWidgetModelRule $rule,
        MagentoWidgetHelperConditions $conditionsHelper,
        array $data = []
    ) {
        $this->_httpContext = $httpContext;
        parent::__construct(
            $context,
            $productCollectionFactory,
            $catalogProductVisibility,
            $httpContext,
            $sqlBuilder,
            $rule,
            $conditionsHelper,
            $data
        );
    }

    protected function _construct()
    {
        parent::_construct();
        // Tested, but does not work
        // $this->setTemplate('Vendor_Namespace::product/widget/content/grid.phtml');
    }

    public function isCustomerLoggedIn()
    {
        return $this->_httpContext->getValue(MagentoCustomerModelContext::CONTEXT_AUTH);
    }
}

├── Block
│   └── Product
│       └── ProductsList.php
├── etc
│   ├── di.xml
│   └── module.xml
├── registration.php
└── view
    └── frontend
        └── templates
            └── product
                └── widget
                    └── content
                        └── grid.phtml

<?xml version="1.0"?>
<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:ObjectManager/etc/config.xsd">
  <preference for="MagentoCatalogWidgetBlockProductProductsList" type="YOUR_NAMESPACECatalogWidgetBlockProductProductsList" />
</config>

<?php

namespace YOUR_NAMESPACECatalogWidgetBlockProduct;

class ProductsList extends MagentoCatalogWidgetBlockProductProductsList
{
    public function myTest()
    {
        die('Hello World !');
    }
}

?>

vendor/magento/module-catalog-widget/view/frontend/templates/product/widget/content/grid.phtml

print_r(get_class_methods($block));

<?xml version="1.0"?>
<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:ObjectManager/etc/config.xsd">
<preference for="MagentoCatalogWidgetBlockProductProductsList" type="VendorNamespaceBlockProductProductsList" />
</config>

<?php

namespace YOUR_NAMESPACECatalogWidgetBlockProduct;

class ProductsList extends MagentoCatalogWidgetBlockProductProductsList
{
  public function customMethod()
  {
    //do your action here
   }
}

?>

vendor/magento/module-catalog-widget/view/frontend/templates/product/widget/content/grid.phtml

app/code/YOUR_NAMESPACE/CatalogWidget/view/frontend/templates/product/widget/content/grid.phtml