I have created a web service API and it's architecture is such that the server requires a client to sign the request along with a secret key assigned to it (signature is always different between multiple requests).
Server matches the client's signature with its own computed signature. If they are a match then the server returns the response.
I am wondering if a client should check the response coming back from the server to see if it's from the same application to which the request was made.
Is any kind of attack possible between HTTP request and HTTP response?
It depends. There are a few types of web service APIs out there. Some need strict security other might not. You could have a few types of APIs:
(1) completely opened API. Say you have a blog where you post about writing RESTful services and clients. You host a complete working REST service based on one of your posts so that people give it a spin. You don't care who calls your service, the service returns some dummy data etc. It's just a demo, a toy, no security here, no request signing, nada. It's just plain HTTP calls.
(2) service with an API key. Say you have a web service and you want to know who calls it. This kind of service needs a pre-registration and each client who wants to call your service needs to register and obtain a key first. Do note that the registration is not about authentication or authorization, you just want to know who's using your API (e.g. what business sector they operate in, how many clients they have, why are they using your API for etc) so that you later make some analysis of your own and take some (marketing maybe) decisions of some sort later on based on the data you get back.
There is nothing secret about this API key. It's just an UUID of some sort, the most basic way of differentiating between calls. This again involves only plain HTTP calls with the key as an additional request parameter.
(3) service with an API key and a secret key. This is similar to number (2) but you need to absolutely make sure that the calls are coming from the client that presents some API key. You need this because you probably want to bill the client for how much they have used your service. You want to make sure the calls actually come from that client and not someone ill intentioned that maybe wants to overcharge the client's bill.
So the client uses it's key for identification and a signature of the request with the secret key to actually vouch for it's identity. This again can be plain HTTP calls with the key and signature as additional request parameters.
(4) data "tampered-safe" web services. For numbers (1), (2) and (3) above I haven't considered any message security issues because most APIs don't need it. What's exchanged isn't confidential and not all that important to protect. But sometimes although the data isn't confidential you need to make sure it wasn't tampered with during transit.
Say you are the owner of a shop that builds some product and you want to advertise your product on some partner web sites. You expose a service with the product details and your partners just use this data to display your product details on their sites. Everybody knows what products you are building so you don't need to hide that, but you are paranoid about your competition trying to ruin you so you want to avoid them intercepting the request and multiplying by 10 all your prices in the responses of your result just to scare potential buyers away.
Number (3) above, although uses the signing part as a way to prove the identity of the caller, also assures the request was not tampered with (server will reject the request if the signature does not match). So if you need to assure an original response you can also sign the response.
For this, the service can provide the client with an API key and two secret keys. One secret key is used by the client to sign their requests while the second secret key is used by the client to verify the signature of the response (using an unique secret key for the server isn't all that safe so the server emits a server secret key specific to each client).
But this has a weak point: you would need to trust your partners that they will indeed validate the response signature before displaying the information on the site and not just bluntly display it. Paranoid as you are you want to protect against this and for this you need HTTPS.
As @SilverlightFox mentioned this proves the validity of the response. The data was not tempered with because it's encrypted. The client does not need to have an extra step to verify the response signature because that verification is already done at a lower (transport) level.
(5) secure services. And now we reach the last type of service where the data is actually confidential. HTTPS is a must for these services. HTTPS ensures the data remains confidential, that it isn't tempered in transit, identifies the server and can also identity the client if client side certificates are used.
So, in conclusion, it depends on what type of service you have.
Make the request over HTTPS to ensure the validity of the response.
This will ensure your data is not vulnerable to a MITM attack. Rolling your own untested encryption/hashing methods is a sure way to open up your application to attack, so you should use TLS/SSL which means that you should connect to your web service API over HTTPS. TLS is the proven and secure way to ensure the response is coming from the application that the request was made to.