ClientResponse's toEntity method converts it to a ResponseEntity<Mono> object. So I came up with the following.

public ResponseEntity<String> doGet2(HttpServletRequest request, URI uri) {
    ClientResponse clientResponse = webClient.get()
            .uri(uri.toASCIIString())
            .headers(headers -> headers.putAll(processRequest(request)))
            .exchange()
            .block();

    return clientResponse.toEntity(String.class).block();
}

I did it in the following way:

   public ResponseEntity<TestClass> getTestDetails() {
        return webClientBuilder.build()
                .get()
                .uri("http://localhost:9090/test")
                .headers(httpHeaders -> {
                    httpHeaders.add(Constants.ACCEPT, Constants.APPLICATION_JSON);
                    })
                .retrieve()
                .toEntity(TestClass.class)
                .block();
    }

You can simply use the Reactor's map operator to modify the headers:

return webClient
        .get()
        .uri("http://localhost:8088/externalService")
        .accept(MediaType.APPLICATION_JSON)
        .retrieve()
        .toEntity(Data[].class)
        .map(responseEntity ->  responseEntity.getHeaders().add("header", "header-value");

Alternatively, you can use .handle operator in order to provide response processing:

.handle((responseEntity, sink) -> {
        if(!isValid(responseEntity)){
            sink.error(new InvalidResponseException());
        } else if (isOk(responseEntity))
            sink.next(responseEntity);
        }
        else {
            //just ignore element
        }
    })