I think the main problem is (probably) that you are returning a promise from within a function which you call (recursively) from within the setTimeout callback, meaning the promise never makes it to the abc function.

When you checkAvailability(options) from within the setTimeout, it returns a promise, but that promise doesn’t go further up (because you don’t do anything with the returned value from the function when you call it for the second or third time).

I would probably go about this differently anyway, but I think that’s where your main problem with the unhandled promise rejection is.

I hope this helps.

Edit: I had a look at it and here is roughly how I would do it:

let https = require('https');
const CONST = {
    MEDIA_NAME: 'twitter',
    CHAR_LIMIT: 256,
    SKIPPABLE_ERR_CODES: [
        187, // Status is a duplicate
        324, // Uploaded media issue with size/validation
    ],
    DOWNLOAD_ATTEMPT_LIMIT: 3,
    AVAILABILITY_ATTEMPT_LIMIT: 3,
    MEDIA_AVAILABLE: false,
    ATTEMPTS: 0
};
async function checkAvailability(options) {
    return new Promise(async (success, failure) => {
        while (!CONST.MEDIA_AVAILABLE || CONST.ATTEMPTS < CONST.AVAILABILITY_ATTEMPT_LIMIT) {
function httpsRequest(params) {
                return new Promise(function (resolve, reject) {
                    console.log('attempting...')
                    let req = https.request(params)
                    req.on('response', (res) => {
                        if (res.statusCode == 200) {
                            console.log('found')
                            CONST.MEDIA_AVAILABLE = true;
                            resolve('success')
                        } else {
                            console.log('attempting again')
                            reject(res.statusCode)
                        }
                    })
                    req.on('error', (e) => {
                        reject(`problem with request: ${e.message}`)
                    });
                    req.end()
                })
            }
try {
                CONST.ATTEMPTS++
                var req = await httpsRequest(options)
            } catch (e) {
                console.log(e)
            }
if (req === 'success') {
                success('success')
                return
                // return is not strictly necessary since the MEDIA_AVAILABLE is now true
                // and the rest of the function does not really do anything
            }
await new Promise(resolve => { setTimeout(resolve, 3000) })
}
        failure('could not find image');
    });
}
async function abc() {
    // let options = {
    //     method: 'HEAD',
    //     host: 'blog.orange.es',
    //     path: '/roducto/ok-google-quiero-hablar-con-orange/'
    // };
let options = {
        method: 'HEAD',
        host: 'xnodejs.org',
        path: '/en/'
    }
try {
        var message = await checkAvailability(options)
    } catch (e) {
        console.log(e)
    }
if (typeof message !== "undefined") {
        console.log(message)
    }
}
abc()