Open up Bitcoin Core and find the transaction. Copy the transaction id without the -000 or anything like that on the end of it. Sometimes Bitcoin Core will tack that on to the end of a transaction id.

Then open up Help > Debug Window and go to the console tab.

Type into the box
where <txid> is the transaction id you copied earlier without the -000 on the end.

You should get a bunch of hex characters as output.

Copy all of it and go to https://blockchain.info/pushtx and https://live.blockcypher.com/btc/pushtx/. In the giant box on both those webpages, paste the hex characters you copied earlier. Then click submit or broadcast transaction and it should broadcast the transaction to the network. If there is any error, you will get a notification on the page. Then you can check the block explorers again to see if the transaction is there.

That is not necessary.

Since the transaction is already created, you should be able to pull the raw hex for it from Bitcoin Core. Get the raw hex and push it through various transaction broadcasting services like the ones at blockchain.info, blocktrail, and blockcypher. You can also try to push it through Bitcoin Core.

That would be the wallet keeping the transaction in its log, would be nice to know why it failed, but anyways, every (proper) wallet has a resync option for this. Its kind of a pain for people using Bitcoin core. Its not a very good wallet for every day uses imo.

I'm pretty new at solving issues, but I also use Bitcoin Core to handle all of my business.
Depending on the web-wallet you sent your coins to it could be a potential problem on their end, however it's entirely possible that your coins are still sitting in your original wallet and need to be Re-Scanned

I had this occur to me once or twice on Bitcoin Core, but it eventually took care of itself after a few hours. Some users have let me know, however that in some cases manual-re scanning may be required.

The TXID does not seem to exist and a transaction for .4BTC does not even exist for that address;
https://www.blocktrail.com/BTC/address/13Y5QidGQzG5QEgytZyQUvQbxDsxdLrLAj

So i'm guessing the broadcast crapped out, not sure if you did a mistake or something but the transaction does not exist so you should be able to retry it already.