user52815: There are plenty of APIs for pushing to the Bitcoin network, like Smartbit's:
https://www.smartbit.com.au/api
However, the only API I found for Bitcoin Cash was Blocktrail's:
https://www.blocktrail.com/api/docs/lang/nodejs#payments_overview
And they do not seem to have a basic push endpoint.

user87240: All the websites listed in MCCCS' answer are no longer operational.
Instead, you can use https://rest.bitcoin.com/#/rawtransactions/sendRawTransactionSingle and paste in your raw transaction after you press the "try it" button.
Alternatively, you can use this command: curl -X GET "https://rest.bitcoin.com/v2/rawtransactions/sendRawTransaction/{your_raw_tx_here}" -H "accept: */*", replacing {your_raw_tx_here} by your raw transaction.
Update:
You can now use this awesome tool: https://bch.info/en/tools/broadcast to broadcast Bitcoin Cash transactions.

user38618: I have listed a lot of Bitcoin Cash block explorers here, and most of them are Insight block explorers so I thought they should have Insight API. I checked all of them and the ones with API are:

Bitcoin.com: https://cashexplorer.bitcoin.com/insight-api/tx/send
BlockDozer: https://blockdozer.com/insight-api/tx/send
Blockexplorer: https://bitcoincash.blockexplorer.com/api/tx/send


52815: <p>There are plenty of APIs for pushing to the Bitcoin network, like Smartbit's:</p>

<p><a href="https://www.smartbit.com.au/api" rel="nofollow noreferrer">https://www.smartbit.com.au/api</a></p>

<p>However, the only API I found for Bitcoin Cash was Blocktrail's:</p>

<p><a href="https://www.blocktrail.com/api/docs/lang/nodejs#payments_overview" rel="nofollow noreferrer">https://www.blocktrail.com/api/docs/lang/nodejs#payments_overview</a></p>

<p>And they do not seem to have a basic push endpoint.</p>
 87240: <p>All the websites listed in MCCCS' answer are no longer operational.</p>
<p>Instead, you can use <a href="https://rest.bitcoin.com/#/rawtransactions/sendRawTransactionSingle" rel="nofollow noreferrer">https://rest.bitcoin.com/#/rawtransactions/sendRawTransactionSingle</a> and paste in your raw transaction after you press the &quot;try it&quot; button.</p>
<p>Alternatively, you can use this command: <code>curl -X GET &quot;https://rest.bitcoin.com/v2/rawtransactions/sendRawTransaction/{your_raw_tx_here}&quot; -H &quot;accept: */*&quot;</code>, replacing <code>{your_raw_tx_here}</code> by your raw transaction.</p>
<h2>Update:</h2>
<p>You can now use this awesome tool: <a href="https://bch.info/en/tools/broadcast" rel="nofollow noreferrer">https://bch.info/en/tools/broadcast</a> to broadcast Bitcoin Cash transactions.</p>
 38618: <p>I have listed a lot of Bitcoin Cash block explorers <a href="https://bitcoin.stackexchange.com/a/57449/38618">here</a>, and most of them are Insight block explorers so I thought they should have <a href="https://github.com/bitpay/insight-api" rel="nofollow noreferrer">Insight API</a>. I checked all of them and the ones with API are:</p>

<ul>
<li><p>Bitcoin.com: <a href="https://cashexplorer.bitcoin.com/insight-api/tx/send" rel="nofollow noreferrer">https://cashexplorer.bitcoin.com/insight-api/tx/send</a></p></li>
<li><p>BlockDozer: <a href="https://blockdozer.com/insight-api/tx/send" rel="nofollow noreferrer">https://blockdozer.com/insight-api/tx/send</a></p></li>
<li><p>Blockexplorer: <a href="https://bitcoincash.blockexplorer.com/api/tx/send" rel="nofollow noreferrer">https://bitcoincash.blockexplorer.com/api/tx/send</a></p></li>
</ul>
