Firstly, we have to create a custom module.

Then, we have to declare a file named events.xml to catch an event that takes place after a product is added to the cart.

Create events.xml file at the following path Magemonkeys/CustomPriceInCart/etc/frontend/events.xml

<?xml version="1.0"?>
<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:Event/etc/events.xsd">
	<event name="checkout_cart_product_add_after">
		<observer name="CustomPriceInCart" instance="MagemonkeysCustomPriceInCartObserverCustomPriceInCart" />
	</event>
</config>

The next step is to create the observer to make changes to the price.

Create CustomPriceInCart.php file at the following path Magemonkeys/CustomPriceInCart/Observer/CustomPriceInCart.php

<?php
namespace MagemonkeysCustomPriceInCartObserver;
use MagentoFrameworkEventObserverInterface;
use MagentoFrameworkAppRequestInterface;

class CustomPriceInCart implements ObserverInterface
{
	public function execute(MagentoFrameworkEventObserver $observer)
	{
		$item = $observer->getEvent()->getData('quote_item');
		$item = ($item->getParentItem() ? $item->getParentItem() : $item);
		// here your custom price goes
		$customPrice = 101;
		$item->setCustomPrice($customPrice);
		$item->setOriginalCustomPrice($customPrice);
		$item->getProduct()->setIsSuperMode(true);
	}
}

By turning on super mode to the product through function setIsSuperMode(true), we can stop the system generating the price.

Then we can set the custom prices for products in the cart.